The best way to get around town is via the subway and bus. Cabs and car services are fine, but they do get stuck in traffic, and for commuting to/from college campuses, you would be fine on public transit since it would not be too early or too late. For late nights, a cab is a good thing, but during the day, it's going to cost you in time and money. New York ia expensive enough, without adding the burden of needless cabs to your grad school budget for commuting. And, unlike most New Yorkers, I do drive around town, so I can tell you first hand, that sometimes it is much faster to get around on the subway and/or bus. Plus, you can get an unlimited pass on the subway, a Metro Card, for $81.
The listings in the Financial District are legitimate; however, the Financial District is a ghost town in terms of amenities for daily living, compared to the UES, UWS, and parts of Midtown and Downtown. You lack many things like multiple grocery stores, cleaners, restaurants, etc. that you will find in surrounding neighborhoods, but it's a hike to get to/from places like that. Plus, to Columbia, that's a much longer trip uptown than you would have in areas closer to the two campuses. If you could put up with these headaches, you can get a newer apartment with amenities at a good price. There's a Trader Joe's and Whole Foods at Union Square, so you are not that far from good stores, but it can be a hassle to go on the subway, but you could always grab a cab back home with a larger grocery order.
The best advice for keeping a budget in check is to find a place that you like that's easily afforded, eat as many meals as you can that you prepare yourself (which is also better healthwise), and don't waste money on needless expenditures that seem to be part of the New York scene. This is not to say that you should not have fun in the city, but there are tremendous opportunities for free and/or lower cost things to do that can allow you to splurge on treats every so often. I have had friends from other cities ask me if it's possible to exist in NY on $50 a day and $500 a day, and the answer to both is yes (exclusive of rent, of course), depending upon where one spends their money. Look for leaks in the budget and eliminate them as best you can, since Manhattan is an island where just about everything you buy is imported. If you think of it as you would a small island nation, it helps to conceptualize the associated costs of everything from dry cleaning to food to household items.
If you don't absolutely need a cleaning service, I'd also skip this expense if you are able/inclined to clean the apartment yourself. Depending upon the space and frequency of cleaning, you might be able to save a good deal of money by eliminating this expense, which can be a New Yorkism that catches many off-guard.
One big expense that you do not need in NYC is a vehicle, with high insurance and garage costs. Check into Zipcar (
www.zipcar.com) for hourly/daily rental for those times that you absolutely need a vehicle to go to the suburbs to stock up on supplies at a warehouse club or Target, large purchases, and getaways. Although, with regard to getaways, one can go quite far on $20, to Boston, Philadelphia, and Washington, DC, for example on one of the bus services like Megabus and Bolt bus. They can be fun for day trips and/or weekends away if your school schedule would allow it.
Hope that helps a bit.