I moved there with about $1000. I was fortunate enough to find a place on the Upper West Side (96th) for $700/mo for a room and a shitty 7-3 $10/hr job and was able to ride it out for 9-10 months. BUT I was too broke to afford open mics, which at that time cost about $5 for 5 min, but a lot of times included the requirement to buy a $5-7 drink as well. I heard a few months back that there are 14 Free open mics a week now, so that is promising. Unable to afford an open mic everyday, I'd hit around 3-4 a month, but what really really really helped was watching all the free shows around town in the bars and at UCB. I never paid a dime to see comedy in NYC other than the donation to the bucket of truth (or the bar equivalent) at the end of the show. NO, sorry, I paid $5 to see Eugene Mirman's weekly show 4-5 times.
Anyway, I had to move back because I quit my shitty job in a huff (couldn't find another one) and decided to take a job back home, that should set me up for a more comfortable return to NY in 2013.
I'd hesitantly say to do it anyway, because you're in your early 20's and NY is the greatest. I don't regret a second of the daily pain in the ass it is living in NY with very little or no money.
BUT I'd also agree with everyone else and say to wait and save money and practice as much as you can before going out there. The community is very supportive and you're honestly doing most shows in front of other open mic comics, not "industry" types scouting for Letterman or whatever.