Yes, it can be done and it can be done easily. I average 30 weekend (2-3/4 days) trips every year - and have since 2007, and never one trip I recall where I even sniffed $300. Couch surfing, hostels, sleeping in airports.net are your friends. I do at least two of those every single trip (usually one night airport - the night I arrive since I usually arrive late on Thursday) and the rest in either hostel or couch surfing. I also do the following on every single trip: jazz bar (if one's available in the city), sporting event (doesn't matter which sport, but I try for the big ones first), museum (free), national park if there's one near the city (I have the $80 pass), sculpture garden OR botanical garden (free or generally < $10), culture (free), people watching (free), photography, nightlife (cheap unless you're looking to get drunk). There's just so much information about free activities on the web these days, you're really cheating yourself if you're spending a lot more than $300 for 4 paltry days.
And dude, you're 19, believe me when I say you won't develop expensive tastes and an affinity to blow money overnight in NYC. If you're within sight or smell of $300, you is a lost cause
BTW, I'm a very socially active, 27 year old male. If someone in my shoes in my demographic can do it, you can, too.
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