Well, Columbia is in Morningside Heights, which is next to Harlem but you'd never know it. Harlem used to be a bad neighborhood, and is still a bit seedy, but for the past few years there's been a huge effort to "clean up" the area and make it safer/more pleasant. Obviously you should be careful at night no matter where in NYC you are, but you're safer there now than you would have been about five years ago.
As far as residence goes, to be perfectly honest, it's nearly impossible to get a room on campus for postbacs. They sometimes have a limited (and I emphasize "limited") number of rooms, but most likely you'll be looking for something
near Columbia instead. It can be expensive, but I've seen pretty cheap rent there too if you can find it.
They do have meal plans, but I don't know the details since I'm rarely on campus long enough to eat. They also have a program called "Dining Dollars" that's more freestyle--you start an account on your ID, and put whatever amount you want on it for food. When you go to the cafe/deli/cafeteria, they swipe your card and deduct the amount. You can even add money to it online. By the way, I highly recommend John Jay for lunch or brunch--everything from scrambled eggs to pizza and a beautiful cafeteria to relax in.