Bonus: if you stay at the Y, you can use their fitness equipment for free (not that you'll have the time or anything....)
There are a number of websites that will help you locate cheap digs as well. One that I like is New York Budget Hotels.A good travel agent will also be able to dig out all sorts of good deals -- if you ask the right questions. I personally can recommend Robin Hutnan. She knows EXACTLY what you mean by mega-budget trip. And she seems to know how to get the cheap flights and stuff. She's actually a goldmine of information about all sorts destinations other than New York.
Overview/Introduction
Before You Go
Where to Stay
Shopping!
How to Get Tickets
Choosing Shows
Cheap Tourist Things to Do
Getting Around
Being Safe
Random Tips/Thoughts
The Barb's Broadway Reviews