The most important wonder in a cultural game is Chichen Itza, the 50% golden age boost is HUGE for a cultural game, if you lose that then forget about a game under 300 turns, unless you're really good. Combine CI with freedom finisher and GA's and you will be in a permanent GA for the last 50+ turns, not to mention the more brief golden ages you will generate b4 finishing freedom, i always beeline towards civil service after i've hooked up the luxuries and teched archery.
So far my times haven't been that amazing though, i still need some practice. 3 cultural games so far - 2 were little over 300 and one a few turns under 300, CI made a huge impact in these games. I have a weakness for the bottom side of the tech tree early on(i blame Tabarnak for that

). So far i've always taken physics for ND, but i think that might be a mistake, since it's slowing me down in tech and i cant reach chivalry for Alhambra in time - the AI seems to love that wonder.
I've been almost totally neglecting religion so far(i pretty much don't bother with shrines early on). That might be another mistake which is costing me in the long run, but even if i'd be able to found a religion with the proper beliefs, i still have to consider conversions - this has actually happened already in my games and is quite annoying. Another fact which is very characteristic of my cultural games so far - i don't build over 3 cities.
Four reasons for that:
1)It will slow down the NC which is crucial for reaching civil service in time(usually finish it somewhere in the early/mid 80's if i won't rush the libraries).
2)It pisses off neighbours if you have 4 cities by turn 50, denounciations guaranteed.
3)You also need extra security/workers/luxuries for that last city which means more hammers/gold elsewhere.
4)The more cities you have the slower the policy acquisition.
So far i haven't had a map where i could utilise 4 cities - the luxuries just aren't there and neighbours are usually too close by to tolerate expansionism.