NYC: Suggestions, anyone?

I'm going to be in the greater NYC area for work at some point in the last couple weeks of January.

Given my passion for cities, andd having 17 business days of vacation that need to get used by December 31 I'm sorely tempted to make a vacation out of it.[1][2]

Here's the thing: I've been in New York City (Manhattan, to be specific) all of twice: The first time was when a business partner took us bowling, yes, bowling -- the second was a free afternoon/evening in October 2008 where I literally wandered around with no plans: Train into Grand Central from NJ, walked West 42nd from 2nd to 11th Ave, plus Times Square, dinner at Burger Joint [a well hidden, literal hole in the wall] in Le Parker Meridian and a show, Boeing Boeing. I stumbled upon Carnegie Hall, again literally -- I caught my foot on the sidewalk just out front -- but I didn't have an opportunity to get inside. My reaction was like "Wow... this is Carnegie Hall... I thought it took practice to get here." ;)

I loved my time there.

But since this trip will be in the middle of January rather than late October, I have a feeling that the climate won't really accommodate my typical "walk around aimlessly and stop when something catches my eye" approach to exploring cities.

So, for those who have been: Should I vacate in NYC in January? How many days? Any must see/eat/do attractions that are off the beaten path? As much as I love the cultural gems of Cleveland, I know NYC is regarded as the epicenter: What two or three or four are the musts?

Lastly, having just written a check[3] for $2k in property taxes I am trying to control costs: Any suggestions for doing the city economically are appreciated. [Though I'll try to cash in HHonors Points for the hotel]

Comment or email away...

Lincoln
[1] I learned this year that 12 consecutive days away from the office just isn't healthy for me. I'm not doing this again this year.
[2] I learned over at the Cleveland Orchestra Blog that Mr. Welser-Most will be conducting the Vienna Philharmonics New Years' 2011 concerts. On a whim I did enter the drawing for the privilege of purchasing a ticket -- I have no expectation of having my name pulled, but if it is I may finally have sufficient motivation to (a) get my passport, and (b) visit Vienna -- so until that's definitely ruled out I'm saving a at least a few days.
[3] Actually "typed the number in my bank's online billpayment thingy" and clicked "Pay Now"...but that doesn't sound as impressive.