I definitely want to stop by Little Italy.
Little Italy is pretty much nonexistent now. Chinatown has almost completely encroached upon them. Not really worth going down there if that's the only thing you're looking to check out.
If you're a craft beer nerd definitely hit up Rattle N Hum. Great beer selection in midtown east and they're bar food is pretty decent too. Also my favorite bars in the city are Blind Tiger and Valhalla .. just to throw out a couple of bar recs.
For eating near your hotel I'd rec Islands Burgers and Shakes for a quick lunch spot. Good burgers and sandwiches. Also further up 9th ave is Kashkaval which is a cool little spot for dinner.
Other than that man there's thousands of places to eat, I would get it down to an area of the city and type of cuisine then get onto Yelp and see which places have good reviews.
If you're in Brooklyn definitely check out Fette Sau. Best bbq in NYC.