If you had ONLY 2 DAYS to eat in NYC....on the cheapish...
if you're a carnivore, and coming from an area that's deprived in the smoked meat food group, i'd highly recommend a visit to Katz's. Lower East Side, corner Houston and Ludlow (just a bit off 1st avenue/Houston). Best pastrami i've ever had. corned beef only so-so, but that pastrami...whatever you do, order at the counter, not from a waitress. if you get the pastrami, tip (ahead of time, as you're ordering) a dollar and the counterman will slice off a few extra slices for you to nosh on while you wait for the sandwich.
If you are going really cheap you might find Katzs pretty expensive, a sandwich and a kinish will set you back around $20 bucks.
For real cheap I'd go to Dumpling House on Eldridge off of Broome - best fried dumplings I've ever had and an unbelievable tuna sandwich on sesame panckae - never had anything like it; will set you back maybe $5. I'd then go to the "cart" on 53rd and 6th for a $5 feast of great hallal food.
For great Itallian I'd hit up Max or Franks in the East village. Max is on B at 3rd and Frank is on 2nd between 9th and 10th (I think). Both have a great East Village feel and won't set you back much at all.
For lunch one day stop at Greys or Kings Papya (located all over the city) $5 will get you a couple hot dogs, fries and a drink. Really a great NYC experience.
You said "cheapish," which by your qualifications "not high end" and "no reservations" (and the fact that it's Manhattan) I've chosen to interpret as "under $40 per person for dinner." With that range, my first choice for my own purposes would be Quartino on Bleecker St. btw. Lafayette and the Bowery. Simple menu, nothing too fancy, but first-quality fresh and organic ingredients, a wonderful cheap wine list (all by the half-bottle), a laid-back only-in-Manhattan feel, so far off the beaten track that you'll have no trouble getting in without a reservation, and a cute garden if the weather's nice.
On the corner of Clinton St. and Houston in the LES (Lower East Side) is Clinton Restaurant and they have the best cuban sammiches.
It's really a Dominican place, and it's not "real" cuban sandwiches, but it's damn good for under $6. Just walk in and ask for a cubano.
You'll also be around plenty of bars and a fun part of town to walk it off. Closest train is the V or F to 2nd Ave.
I went to eat at Yokocho around the corner from St. Marks (next to Around the Clock on I think 9th St. and 3 Ave) and we ate like champs for $20/person including tip. No booze though. They wouldn't let me drink since it was a Monday (long weekend of heavy drinking if you must know).
I second Max—great Italian and very reasonable. I usually go with the pasta of the day.
Caracas on 7th St just east of First Ave. They have phenomenal arepas. Try the chocolate milkshake, too.
Westville on West 10th b/w Bleeker and 4th St. Standard American dishes—good sides and desserts.
Cafecito on Avenue C b/w 11th and 12th Sts. Great Cuban fare and cocktails.