Craig's List is a place where people can list all sorts of things. It is probably your best bet for finding an apartment without going through an agent or broker (which would be a real education regarding our cost of living). Cynthetiq had a good suggestion with the Village Voice. That's a free paper here in the city, and it has apartment listings. However, check it online, as the listings there are updated every day. I think this requires some small fee, but it is worth it. The print version only comes out once a week, and by then the online people have worked it over pretty well. That reminds me about something else that people from other cities aren't always used to... If you are serious about a place, you can't really think about it for too long. Apartments at good prices in the area you are talking about can be snatched up in one day.
I'll ask around with some of my friends about sublets for the summer. Late June is a little tough since most students are already gone, but I'll see what I can do. You may have to consider living in one of the other boroughs or upper Manhattan (above Harlem). That is what a lot of students end up doing. The commute to mid-town isn't so bad if you live near a subway stop that goes there.
I don't mean to make it sound like what you want is unrealistic here - if you have your money in your hand and can act quickly, you should be able to find a place to live. One more thing about Lincoln Center - you do know that Lincoln Center consists of 13 different organizations, right? You could ask each one of them if they might need an intern or some sort of summer work. That would include (I can't remember the one):
Lincoln Center, Inc.
Jazz at Lincoln Center
Metropolitan Opera
NY City Opera
NY City Ballet
New York Philharmonic
Lincoln Center Chamber Music Society
Performing Arts Library
School of American Ballet
Juilliard School (212)799-5000
Vivian Beaumont Theater
Walter Reade Theater
You could also try:
Carnegie Hall
Merkin Hall
Norton Symphony Space
Manhattan School of Music
New York Pops Orchestra
Mostly Mozart Festival
American Composer's Orchestra
Orchestra of St. Luke's
You should be able to find the phone numbers for these organizations on the internet pretty easily (I put Juilliard's number on here for you, since it is the only one I know off the top of my head). Cynthetiq might be right about it being too late, but you could always call them up and explain your situation. Someone may have some sort of job you could do, even if it was office work - that would be a foot in the door for next year. That is how I ended up with an internship at the American Chemistry Council a few years ago.