From the New York Times, December 30, 2010: A New York City Weekend for $100
Of course the above article only scratches the surface. For cheap food, you can't beat Chinatown. And don't forget Brooklyn's Chinatown on Eighth Avenue in Brooklyn, which is even cheaper than Manhattan Chinatown.
Whether you know anything about art or not, I recommend walking around Soho or Tribeca on a Sunday afternoon. Many individual artists and galleries display their work on the sidewalks. I have seen art on the sidewalk or in gallery windows that was amazing, and I don't use the word amazing lightly. And anyone interested in architecture will find an abundance of notable buildings to look at in Manhattan.
Just to relax, I recommend walking the river front promenade in the Battery Park City Area. Visit the children's playground. Look at the sculptures of the frogs and turtles and poetry inscriptions. Drop in on the Irish Hunger Memorial and the Holocaust Museum. Walk over to Ground Zero. Take a stroll to Park51. One thing that I have always wanted to do but never have, is to walk across the Brooklyn Bridge.
If you like to take photographs, there is no end of things to take pictures of. Definitely take in a ferry ride of some sort, preferably one that goes near the Statue of Liberty. It is rather dramatic when seen close up from a boat. But also, New York Harbor is beautiful when seen from a boat. It also yields one of the best views of the skyline. If the weather is good, a walk in Central Park can be stimulating too.
Visit locations in NY that have been used as scenes/locations in movies or cited in works of literature. Then of course, there is a laundry list of famous, old, or trendy saloons to wet your whistle afterwards.