university libraries aren't open to the public.
I'd start with New York City's public library system. Find a librarian to help you get access to the various electronic databases, which includes Academic Search Premier and a bunch of others. If you need a specific journal article, print out the abstract or citation that you found on line, and bring it with you to the library. You can often find similar and/or more up to date articles for free with the help of the librarian. If access to the article isn't available through the public library, you might be able to get METRO access to a university library or private special collection: http://metro.org/referral-cards/. A librarian could also try to ILL you a copy of the article, but it takes some time.
Public university libraries are almost always open to the public...that's SUNY and CUNY. If you are lucky, you can get a daily guest password for the computer databases, but YMMV. I'd call around to a number of public universities and ask about guest policies.