I really don't understand how anyone could finish this in under 200 turns.
I don't know how you even explored the whole map in less than 150 moves. Did y'all peek at the map with the developer toolkit or something?
When I tried rushing Rome with 2 composite bows and a pathfinder, I was swarmed by barbarians and killed just as I was about to take the city. (so I quit and started over)
I finished the second attempt at about turn 400. I stalled at Japan, (click, click, click) waiting to declare war until reinforcements to make their way all the way from Moson Kahni. By this time I had crossbows, knights, and muskets. I captured Osaka (and Kyoto and Tokyo) without much difficulty, but I lost my pathfinder-crossbows in the first battle, which slowed things down considerably moving through hills and forests.
Liberating Dublin to bring the Celts back to life turned out to be a good move. If I captured a fairly high population city, I could set it ablaze, sell a building or two, upgrade any of my units that could be upgraded, then give the city to Boudicca. Let her finish burning it down instead of me, so I don't get the happiness penalty. My happiness never dipped negative, but it got close a few times.
I puppeted almost all the cities I took; razed a few, gave away 2 or 3, and liberated all the CS's. I only annexed Osaka, so I could buy new units there. By the end I had a vast turquoise-colored puppet empire, and was generating about 600 GPT (over 900 during golden ages) from all the trading posts. I had more gold than I could spend, but I don't think buying more units would have sped things up much because of the travel times. I suppose I could have embarked a bunch of infantries and sent them unescorted across the sea to invade the Zulus while I was still fighting the Mayans... most of them would have made it across.
Once I researched Astronomy (and especially once I researched Steam Power) I could build/buy super units in my capital (Heroic Epic + Brandenberg Gate) and resupply my army across the ocean rather than snaking my way around by land. I bought all my naval units at Osaka, because the only coastal city I settled myself was blocked by ice.
Comanche Raiders are more useful than I expected. Especially for squashing barbarian encampments, or picking-off damaged enemy units before they have a chance to heal. They also make good spotters for artillery (towards the end of the game I was mostly using spies for spotters)