Central Park Cyclist here, so here are some shops that I think are worth checking out:

You'll have to Google to find the website and local but definitely worth checking them out.

Cadence
Gotham Bikes
SBR (sbrshop.com - great central park tri-team)
Bicycle Renaissance
Toga (a few locations)
NYC Velo
Sids Bikes


In terms of the OP's request, let's try to remember the NYC today is NOT the nyc of 5-10 years ago. If you are a bicycle shop in manhattan, you know it's all "boutique" esque. That means everything is sticker priced, there is no more bargain and barter in such a high profile city. You WILL however find small mom-n-pop bicycle repair shops who may point you in the right direction. These are the bike shops that repair "delivery bikes". This one is my home shop:

Master Bike - 77th and Broadway

There are many more small bicycle shops that don't get visibility on the web simply because they don't sell sticker priced bikes. I know for a fact that Master Bike can order in a bicycle but hardly have the space to have a showroom of them. It's more of a repair shop, which would be your BEST BET in finding a cheap deal. Other then that, I'd recommend craigslist where you'll have your best chance at finding something affordable.

Hope that helps...
-SD


PS - boathouse in central park gets some attention, but the weekends is full of blazer adorned boat shoe brunchers and the atypical couples rowing boats in the lake. I'd camp out outside of sheep meadow or the lawn adjacent to Metro Museum of Art (east 60s?) across the main central park road.