This game offered some complex scoring mechanism where the challenge lied not in winning the game but getting a higher score than other human players, so I played it despite the low difficulty of Prince. The game itself was even easier than anticipated and a bit tedious too, as it was on a large map and the scoring called for a lot of unit movement (both missionaries and normal units). I will offer you only a short report, as the game itself turned out not to be very exciting - contrary to most of my games, the AIs were extremely tame this time.
The main goal of course was to achieve more than 150 points in each of the three categories. For the Farmer, this meant founding cities so that they would be able to grow to size 25, and adopting hereditary rule, state property, bureaucracy, and organized religion as soon as they were available and sticking to them. The score for the health resources at 1500AD I ignored more or less and didn't tailor my game towards it, as it was too insignificant.
For the Priest, it was important to found a lot of religions and spread the state religion as much as possible. To found lots of religions and to build their shrines, great prophets would be needed, so some early religious wonders would have to be built.
For the Soldier, capturing (and growing) opponent's cities was the key, as well as seeing to it that one unit would fight a lot and survive for the xp score. Capturing (and keeping) a lot of cities would be maximized by going for a domination victory, as I figured that this way, I would be able to get more than the 20 points those players get who win a diplomatic victory with less than 40% of the world's population.
We did not start with Mysticism, so I decided to ignore Buddhism and Hinduism and go for the later religions instead.
What I did not take into account though was that I am the god of tribal villages in my games. 
Somehow I always get at least one tech from a hut, and this game was no disappointment in that regard.
So when I received Mysticism from a hut in 3840BC, I immediately switched research from Animal Husbandry to Polytheism,
trying to found Hinduism. Going for Hinduism is a much safer bet than Buddhism, and indeed Buddhism was founded by
somebody else quite early (Isabella, as it turned out), but I managed to get Hinduism in 3320BC.
After completing my research on Animal Husbandry, I went for Bronze Working next. I planned to build the Oracle later on for the great prophet points, and wanted to take Metal Casting as my free tech for which I needed Bronze Working anyway, so I wanted to see early if I had copper nearby, in which case I could skip Archery. And indeed, copper was right at the capital, which made this game even easier! I never built a single archer during the whole game, only axemen instead, which stopped the raging barbs dead cold.
This also made Moscow a production powerhouse, so after building a second scout, a warrior for MP duty and a worker, I built Stonehenge, then later the Oracle. I quickly found Qin, later Isabella and Saladin on my continent. My axes easily paved the way for my settlers to expand my empire, so the early game was quite straightforward. Here's a shot of my core in 500AD:
I built a lot of wonders throughout the game: The great library, the great lighthouse, the hanging gardens, spiral minaret,
Versailles, Taj Mahal, Statue of Liberty, and more. I also discovered all relevant techs first: Music, Liberalism, Economics,
Physics etc. Both wonders and techs resulted in a lot (and I mean A LOT) of great people, which were
mainly used for shrines, academies, and golden ages - lots of golden ages in fact, one with the Taj Mahal, and three(!)
with great people. 
I'll split the rest of my report into the three scoring categories Farmer, Priest, and Soldier.
I founded cities wide-spaced so they could grow to size 25 eventually. Some trade deals for happiness resources allowed me to grow three cities to size 15 at 1500AD, Moscow, St. Petersburg, and Novgorod.
I had 11 health resources connected in my territory at 1500AD, and had founded 9 cities overall at game's end, five of them at size 25. More pure Russian size 25 cities would have been possible if I had razed more cities, but I had been too lazy to do this.
I adopted the scoring civics as soon as I had researched the techs, and never revolted back. That led to a suboptimal game later on, as I normally would have switched to war-time or research civics instead, but that didn't matter on Prince.
After I got lucky with getting Mysticism from a hut which had encouraged me to found Hinduism as my state relgion, multiple great prophets helped me with founding Christianity and Islam as well. Shrines were constructed for all three, albeit the last as late as 1924AD. Financially, only the Hindu shrine was important to me, as the other two religions were founded not in Moscow. Isabella founded Buddhism, Saladin Judaism and I forgot who got the others.
I made sure to build missionaries whenever I could, even during war-time my military-producing cities would pause to build a missionary or two if the option became available. Isabella and Saladin I could not convert as they had founded their own religions, but Qin converted to Hinduism even before I could send a missionary to him, and I later managed to convert Hatshepsut as well. Not very much, but when my caravels found the other civs, other religions had already been established there. Nevertheless, most of my missionaries were ferried over by caravels; later I used airports to bring them to the continents I was conquering. Unfortunately, these missionaries meant quite a bit of tedium. I could have built more, but I was too lazy to manually switch a city to build one as soon as one was consumed, and instead waited for the next city build prompt instead, which probably cost me 20-30 priest points (multiplied by the Great Tsar bonus).
By 1000AD, 11 cities had the Hindu religion; at game's end, it were 77. At 1500AD, my religions had 21% influence.
The soldier in me received the most attention. In BC times, an axeman, battle-hardened already by skirmishes with barbarians, met a Spanish archer/archer/settler pair on the way to a nice city site. So I called in another axe that was scouting the area, waited some turns until I figured the city would have grown to size 2, and declared war in 325BC.
My two axes won their fights, and I captured Cordoba - only to see that a barbarian archer was just behind the city,
who in turn attacked and killed my wounded axe, taking the city! 
My second axe managed to retake Cordoba, though.
After that, bringing in reinforcements from home took a long time without roads, so I managed to take a second city from Spain only in 540AD, after the 500AD scoring deadline. But that city north of Cordoba was very important for the Russian empire, as it had horses! I regarded these as critical to have, for my cossacks later.
I made peace afterwards and built up my empire until 1570AD. By then, I had produced my first few cossacks and decided it was time for Qin to go, as he was the second most advanced civ on the planet (behind me of course).
As you can see, I only had four cossacks and three catapults ready, but that was more than enough to declare war on a Prince AI. I quickly captured Shanghai, then Guanghzou, then the rest of pathetic China. I met only longbows and macemen as resistance, but not only that - the AI had nearly no troops to make counterattacks with, so that war was a joke. Speed was only limited by availability of reinforcements and healing times. China was destroyed in 1720AD.
So as soon as my cossacks were healed and moved east, I declared war on Isabella again in 1735AD. She offered even less
resistance, but then losing two cities and lots of units by 500AD must have hampered her overall growth a bit.
I made peace with her in 1826AD, at what time she only had one city on an island
left.
I waited until 1834AD until declaring war on Saladin, because he had been peacefully expanding and teching until now, so I expected a lot more resistance from him. That didn't happen though, he just fell over - I have no idea what happened there; maybe it was because of Prince level? He had rifles and cavs, but not a lot of them. Here's a shot from 1834AD (some of Saladin's cities are not showing due to some bad intel on my part):
So while I was mopping up Saladin, I prepared a fleet to invade Huayna next. I had tanks now, and on my way to Huayna made a detour to the island where Isabella and Saladin had a city each, Isa's last in fact, and captured them. I then declared war on Incans in 1922AD. Huayna had a defense pact with France, so actually I declared on both - but that didn't matter. Again, I met no real resistance and made peace with Huanya in 1955AD when his last city was out of reach for me at that moment. While capturing Incan cities on their continent, my navy destroyed some galleons full of units from France, then I invaded France in full force. While finishing them up, my navy prepared to land some modern armor in Azteca.
He only had infantry and artillery, so again it was a breeze. I won a domination victory in 1983AD - rather late, as I had overestimated the AI's power all the time. Here's my most experienced unit at game's end:
I had taken a cossack to go for a high-xp unit instead of a axeman/maceman because of its movement speed. I could have gathered experience with an axeman or maceman much earlier, but it would have later slowed down my conquest when my main attack units were cossacks and tanks. Also, I feared I would lose the axe/mace trying to kill (wounded) riflemen, artillery, or infantry. Now, my bombers would soften up the city garrison, then I would send in some barrage tanks, and then my cossack would have about 98% chance of winning the fight.
Overall, I captured 64 cities after 1500AD, and managed to grow 20 captured cities to size 25. Again, more would have been
possible with some serious dedication to ferrying over workers and irrigating everything. I also could have stopped
warring shortly before domination to convert more cities (Hatty had only two sparts of the spaceship built by then).
Or I could have killed all but one defender in every city on the planet, then taking all cities in one turn to get
more points. Or I could have founded lots of junk cities on the last turn. But...
the game had taken too long as it were already.
