For theatre experiences, The Book of Mormon and Sleep No More are truly incredible. Although, I don't know how much luck you'd have in finding tickets.
I believe that The Book of Mormon is from the creators of South Park, and it is absolutely hilarious. It's about two young Elders that go on their mission to Uganda and all hell breaks loose. I heard they're sold out for a few more months but it never hurts to see if there are some tickets out there.
http://www.bookofmormonbroadway.com/home.php
Best part of the musical:
As for Sleep No More, it's a completely different theatre experience where the audience members put on masks and walk throughout the warehouses that have been meticulously designed to create the set from Shakespeare's Macbeth. You are basically walking around with the actors.
http://sleepnomorenyc.com/
NYC is incredible! Do everything that you can in your short stay - there is ALWAYS something to do or see in NYC.