First time in NY, good cheap eats.
You're right in the middle of Koreatown with its wealth of wonderful, inexpensive restaurants, cafes and bakeries. Walk west along 32nd St. and you'll see a LOT of options. Here's one thread to get you started:
Korean bbq is a little more expensive, but there are probably places where you could, with careful ordering, get out for under $20 / person.
Han Ah Reum supermarket on 32nd just east of B'way sells decent takeout containers of jap chae, maki zushi (vegetarian, Korean style), mung bean patties and ddeok at lunchtime.
Mad for Chicken recently replaced Bon Chon Chicken and might be an interesting choice. Bon Chon served some of the city's finest (Korean style) fried chicken at one point. I haven't tried Mad for Chicken, which says it serves exactly the same chicken as it did in its prior incarnation, but I've heard some good reports.
Tio Pio West is a short walk away should you have a hankering for Peruvian style chicken. Midtown Lunch review here: http://midtownlunch.com/2008/01/24/st...
And if you don't mind a longer walk (you'll need one after gorging on all the good K-town eats ;) Piece of Chicken sells southern-style fried chicken for $1 / piece and decent accompaniments, too. It's on 45th between 8th and 9th. Yelp reviews here: http://www.yelp.com/biz/piece-of-chic... . I personally don't think it's the greatest fried chicken on the planet... but the Baptist ladies in my hometown made a fried chicken snob out of me and hey, it's only a dollar. The fried chicken livers are an interesting side.
Also on 45th St. between 5th and 6th, there's Sukhadia's, which makes Indian chaat and has some sort of good-value steam table buffet at lunchtime. The food is pretty decent and (mostly? or all?) vegetarian. Sukhadia's is actually best known for its Indian sweets, but those might be more of an acquired taste... they're tooth-achingly sweet.
I know you'll run into a few of those ubiquitous-in-NYC halal chicken and rice carts in the area, but I haven't really sampled enough of these to know which ones are good. Hopefully someone who lives in that neighborhood will be able to give you better street food recs.
Wow... what a chicken-centric post this was. I hope you don't get the wrong idea. There is, in addition to chicken, other delicious food to eat in NYC. :) Enjoy!
Edited to add...
On the non-chicken side of the equation, you are also within walking distance of two NYC institutions:
1. Ess-a-Bagel, which is often mentioned in heated Chowhound debates about where to find the best bagels in NYC. The closest location to you is in Murray Hill on 1st Ave. at 21st St. This is a longish walk from where you're staying, but on a pleasant day, it's a good way to see the city.
2. Shake Shack in Madison Square Park at around 24th St. and Madison Ave., one block east of 5th Ave. The waits for burgers and concretes at peak times are legendary / horific, so, if your schedule permits, go at off hours -- i.e. 11 a.m. for lunch or 3 p.m. for a snack.
If you're looking for a street vendor, Hallo Berlin cart at 5th and 54th is my absolute favorite. Delicious wursts of all kinds, german fried potatoes and cabbage. It's amazing. For non-cart but still cheap good Indian food, check out curry hill, just a couple blocks east and south of where you're staying. Tiffin Wallah at 29th and Lexington and Chennai Garden at 28th and Lexington both have amazing veg lunch buffets for $6.95, all you can eat.
These aren't street vendors, but if you're a hot dog aficiando check out Crif Dogs ( http://www.crifdogs.com/ ) - awesome doggies, jazzed up in all kinds of ways like the BLT (pork wrapped in pork...mmm!) and chili dog. For burgers, Corner Bistro ( http://cornerbistro.ypguides.net/ ) is legendary but Burger Joint ( http://parkermeridien.com/eat4.php ) probably has better burgeers (Bistro wins on atmosphere & beer hands down though). Russ & Daughters has awesome bagel sandwiches if you like stuff like smoked salmon, cream cheese, etc ( http://www.russanddaughters.com/about...).
If you're only spending one day. I suggest spending a lot of time in the East village. It is personally my favorite food neighborhood in manhattan. Being a college student I have a good idea for cheap eats.
I would plan an East Village cheap eats trip like so, begin at Artichoke pizza for a square slice. Then stop by next door to Zeppole for cheap desserts like a deep fried oreo or zeppole. I also like their pb & j italian ice. You can then walk over a couple blocks or take the 6 train on stop to St Marks place. As mentioned there is Crif dogs. But there are also a few Yakitori (meat on a stick) places I'm not an expert on them because I've only ate at these places a 2-3 times but I generally felt they are all about the same level. My favorite yakitori is the chicken skin. There are also takoyaki balls (tasty octupus balls) at St Marks I like Otafuku for those. I've never been to Japan but I think this is street food there or at least cheap bar food. I would also stop at Mamouns for a shawarma, hummus, or falafel. To top it off I would go to Chikalicious Dessert Club for some to go desserts. The desserts there aren't that expensive. My favorite things there are the vanilla custard steamed, smores cupcake and the vanilla bean ice cream. But i've only been there once. So there other things maybe good and it would be much easier to share. I'm pretty sure all this can be done with 20 dollars pp as long as you don't order too much and share.
I would also do this stuff at night since all these places close late. Maybe get some cheap eats in chinatown during the day (dollar dumplings, bahn mi, hand pulled noodles, etc) since they close earlier.
P.S. At St Marks place there is a place called Kenkas which has 1.50 beers (sapporo or kirin) in pretty big mugs. The food is kind of expensive and nothing to write home about. But the mugs will do you some damage. I would avoid it though unless you want to drink, I believe there is a 10 dollar min pp. I'm not sure though, I'm usually too drunk when the check comes.
ENJOY! sorry you're limited to just one day. I know you want to limit yourself to 20 dollars but if you truly love food I would take out some extra cash. I use to live 40 mins from manhattan via subway. Now I'm 2 hours away and cannot visit as frequently so I am suffering withdrawal. If I were to move far away from NYC tommorow, I certainly wouldn't hold back on the wallet.