Oh man, such an intense game.
I usually consider myself an Emperor level player, but this Prince game really taught me a few lessons. I almost managed to snatch defeat from the jaws of victory, only to snatch victory back again!
At the end of my first spoiler I didn't have a hugely interesting game to report. I was heading for a semi-peaceful cultural victory, spreading my religions amongst my cities and happily building things and amassing culture.
I discovered CS in 450 AD - and then FORGOT to revolt to Bureau until 760 AD.

In 700 AD I learned Music, but evidently was not the first to it, since I didn't get the free GArtist.

In 1100 AD I learned Philosophy - about 300 years after Taoism was founded.

In 1210 AD I was the first to Liberalism. Yay, a nice positive.

I take Nationalism for free, and after a few turns I finish Drama and Printing Press, where the researching stops at 1370 AD (Replaceable Parts at about 1 beaker per turn).
Ok, so I'm on track for victory. I noticed the Hanging Gardens hadn't been built yet so I built that, shortly followed by Taj Mahal. I'm pretty much keeping up in tech through trade up until the 1800's. I'm estimating my finish date for about 1860 AD.
1824 AD: Aargh! Huayna Capac is sitting just outside my borders with a massive fleet of frigates and galleons loaded with Cavalry, Cannon and a few riflemen. Alarm bells are ringing - he is nearest to a satellite city with only a longbow and a warrior defending it. Things aren't looking good for that city. Time to turn back on the research (Military Tradition) and generate some cash for unit upgrades, and shuffle some units around. I can weather this storm - only 40 or so more turns to victory.
1826 AD: As expected, HC DoWs on me. I have a few muskets for defence, and a whole lot of keshiks left over from my wars with Washington and Isabella. HC lands next to Murcia on the SE coast of my lands. He doesn't attack it immediately, instead pillages some improvements.
1830 AD: Murcia is captured and razed! Uh oh. Thankfully, HC's units took enough damage that the few Keshiks I promoted to Knights are able to mop up. Ok. So that was bad, but I'm not dead, and my designated legendary cities are unaffected.
1834 AD: I discover Military Tradition. Now I'm saving cash to upgrade all my Keshiks and Knights. Satellite cities are producing more cavalry, muskets and catapults. I'm feeling good. Back to investing commerce into culture. I can weather this storm - only 40 or so more turns to victory.
1888 AD: HC has returned. This time off the coast of Washington, which is to become a legendary city. Cavalry, a couple of rifles and Artillery this time! Aargh, stupid stupid. I am officially panicking, so I hit the panic button. Revolution to Nationhood. HC is swarming over my lands pillaging my most valuable towns and villages.
1900 AD: I eventually get the infestation under control, at the expense of a few divisions of cavalry. Revolution back to Free Speech. HC won't take peace for anything, so I figure my best way out of this war is to end the game. I have about 6 Artists waiting in the wings, but my calculations still suggest Washington is about 10 turns from legendary. (I had more artists in the game, but some were bombed earlier in New York).
1908 AD: Karakorum achieves legendary status. My cities are terribly out of sync, since Washington had its commerce production shattered by HC. It gets worse. HC has landed more units next to Washington - now I'm facing cavalry, artillery and SAM infantry! I don't have enough units.
1910 AD: My designated legendary city, Washington, is captured and RAZED! NOOOOOOOOOOOOOO... *sob*. Do I quit? I'm very tempted. But, I have my secondary GP Farm, Santiago by the floodplains east of Karakorum. It has amassed quite a lot of culture running all those artists. I can still do this. I'll need rifling though. I can weather this storm - only 40 or so more turns to victory.
1923 AD: I discover rifling. Riflemen aren't bad, right? Maybe I'll build a few more catapults as well. Back to full-time Culture commerce.
~1940 AD: Hmmm, a bunch of players have built Apollo Program. HC has landed another stack - this time right next to New York - my other legendary city and main GP Farm! I'm now facing Artillery and Tanks!!! Taking New York is unacceptable. I throw all my catapults at it, followed by my cavalry. Surprise! I actually eliminated the whole stack! Some of my catapults even managed to withdraw. new strategy = many many catapults + many many cavalry.
1945 AD: Huayna Capac has built Apollo Program. Some respite perhaps?
1950 AD: I'm holding my breath. HC's navy is keeping my coastal cities pretty much permanently on 0% defense.
1960 AD: I'm turning purple from holding my breath. I'm building nothing but cavalry and catapults.
1966 AD: New York is 1 turn away from Legendary. Time to put Santiago over the edge, so I bomb 8 Artists. Santiago achieves legendary culture!
1967 AD: New York has achieved legendary culture! Woooooo.
Final result: Cultural Victory in 1968 AD.
Lessons learned:
* don't neglect your military!!!!! Ever!!! !
* I guess my diplomacy could have used some more work - HC was Cautious when he DoWed IIRC. I gave in to all outright demands, but occasionally I refused to stop trading with XX when asked.
* Unless you're planning on settling a Great Artist, save the culture bombs until the very end. Who knows, that city you bombed in 1000 AD might not be around at the end.
* Catapults are great. They may not even scratch the top defender, but they still do damage to all those other units. Even if they only take off 5 hp per unit, with enough of them you'll have a stack of tanks down to a level where cavalry can finish them off.