Go to the Met. Temple of Dendor. Done. There's a Cezanne exhibit going on now as well.
Hit Gray's on 72nd, it's a NY thing and is much better than a dirty water hot dog.
There's a Turkish restaurant, Turkuaz, near your hostel. It's damn good but not dirt cheap.
If you're looking for incredible pizza, go to East Harlem to the original Patsy's. For a slice, Joe's. Artichoke is many things, including good, but it's not NY pizza.
On the UES near the Met is an italian sandwich restaurant called Via Quadronno. Great coffee as well.
In the East Village is a panini shop that only sells porchetta panini and plates. It's called, wait for it.. Porchetta.
On the same block (7th street, btwn 1st and A) there is also a cheap lobster roll place, and arepa place, and a few other cheap restaurants. They're all very good, and cheap.
St Marks place has Yaktori places that are good but the midtown versions are generally better. Tuesday night on of the midtown places has $1 skewers. Big line, but cheap beer and food = a good night.