Food: The best cooking school in NYC has a (relatively secret) restaurant that is inexpensive. If you like high cuisine, this is it for we mortals. Reservations and directions: http://www.frenchculinary.com/lecole.htm
Oyster Bar in Grand Central. If you like oysters, here you can compare them from around the world. A rare thing. I go to the counter. http://nymag.com/listings/restaurant/oyster_bar/
McSorley's is kind a tourist attraction, but it depends on the time of day you go. It IS authentic (women are now tolerated). http://www.mcsorleysnewyork.com/
Shopping: Beretta Gallery (http://www.berettagallery.com/index.aspx?m=53&did=355
This link is broken, either the page no longer exists or there is some other issue like a typo.
) has high end italian 'country' clothes that you will not find in Georgetown. I go up to the gunroom, while melady shops amongst the Gulfstream daytripping ladies from Houston.
FAO Schwarz is amazing in the Christmas season. http://www.fao.com
Since you don't have much time, a simple stroll up 5th Ave and down Park Ave will give you plenty of adventure watching people and finding great shops. Duck into a hotel lobby for a comfortable rest and sip of something (yes, coffee and a snack will cost as much as lunch, elsewhere.) Falafel in a pita from a street vendor is a great $4 `snack.