How much is possible to do for a day in NYC?

Here's your problem... The Statue of Liberty/Ellis Island and the WTC are downtown, the Empire State Building and Rockefeller Center are in Midtown and the Museum of Natural History is Uptown. If you have only one single day, the only logical advice would be to either stay in one area of Manhattan (Uptown, Downtown or Midtown) or two (Midtown + either Downtown or Uptown). You can't do all 4 in a single day without it being totally half-assed. For example: Visit the Statue of Liberty and WTC (both downtown) + the Empire State Building and/or Rockefeller Center (both midtown). OR visit the Museum of Natural History (uptown) + the Empire State Building and/or Rockefeller center (both midtown). Mixing Downtown and Uptown will kill your day.