First off, congratulations on moving to one of the city's famously non-existent "neighborhoods" (in that nobody, save for real estate brokers and the glossy lifestyle rags refers to the genearal area to the east of Morningside Park as "South Harlem", not with capital letters in any case).
That said, since you seem to like moving around during the day -- and since you'll get used to jumping into the subway or otherwise traveling considerable distances throughout all hours of the day and night, far more frequently that you would in most other large cities -- you might want to take advantage of the strategic B and D lines, which will put you within 20 minutes shooting distance of the NYPL branch at 5th and 42nd (after a nice stroll through Bryant Park, from where you'll be getting off). Quiet and comparatively decent surroundings, if you don't mind going through bag checks everytime you step out for coffee, and having a security guard come by to rap his knuckles on your desk should you find yourself dozing off. And now open 7 days a week (having sold its soul and naming rights to some zillionaire benefactor or another).
If you can work in coffeeshops (and have a strong battery in your laptop or netbook), there's the Hungarian Pasty Shop (behind Columbia, at Amsterdam and 110th; a nice uphill stroll through Morningside Park); many find it oddly conducive to work (and hacking) despite the cacophanous chatter, and despite (or perhaps because, depending on your point of view) the lack of wifi (or outlets). The espresso is sub-par, but the baked goods are decent and decently priced, and the staff are quite tolerant and permissive.
Those same trains, along with the A and C will also take you down to the Greenwich Village / NYU area, which has a plethora of options on the wifi/coffeeshop front (Think Coffee, Tea Shop, Reggio, and Grounded are all quite reliable).
Outside the coffeeshop mold, there are a lot of other de-facto "study halls" embedded in strategic throughout manhattan: the atria of many larger office builings (basically, open spaces the developers were forced to provide in exchange for air rights and such) have been converted into public lounges essentially; many now have wifi and cheap espresso bars.
And if you're still blowing through your cash, there are even more scenic options, like the courtyard at the MOMA (if you felt like dropping a bit more than the usual coffee bill to visit the museum proper later in the afternoon).
May or may not make up for the near-impossibility of finding decent-size living and work quarters, sans roommates, anywhere near the center proper, but these options do provide a break from the usual office or work-at-home tedium, at least.
"Poverty with a view", we call it here.