Anastasia is supposed to come to broadway in the spring, too, but who knows how long it will last (is it ok to talk about non disney princesses?).
As far as Broadway - Aladdin is the best (energetic and funny) and the New Amsterdam Theater is gorgeous! Lion King was okay (obviously I am in the minority there considering it will be celebrating its 20th anniversary on Broadway next year but I LOVED the movie). Matilda was good but will be gone by the time you get to NYC and Frozen won't be there until 2018. If you aren't able to catch a Broadway show, check New Victory Theater on 42nd Street. They have several excellent shows for families.
My feeling on the Lion King (which I took my 8 year old too just before our Canada cruise) - I'm sure it was incredibly mind blowing 20 years ago, but now, we saw some of the same puppet effects on our cruise!! Though I will admit to getting a little teary when a rhino walked right by us in the opening number.
"What should I do in NYC" is too broad of a question. Basically, anything you can think of, you can probably do in NYC so I feel like more info is needed. Do you like museums? Art? Outside stuff? I'm close so I took the train in and the people next to me were going to a Tim Burton themed brunch. Really, if you can imagine it, you could probably find it in NYC!!
While I personally probably wouldn't do anything on the water before a cruise, I like the Staten Island Ferry. It's free and you go by the Statue of Liberty and Ellis island and it's a lovely view of lower manhattan.
And have back up plans. It was ridiculously grey and gloomy before our cruise so I would've cancelled any plans to go up any buildings (really you couldn't see the tops of most buildings) and I don't know if I would've done sightseeing busses on the rainier days.
The only other thing that's we often do is a quick stop at the
lego store at Rockefeller center. I, personally, hate it, it's small and packed, by my 8 year old loves it.