that's cool if columbia offers close and cheaper housing, but other than that, good luck. Sure, there may be a steal or two in harlem, but finding those are rare. The more uptown you go, past 125th, the area is not necessarily ideal (crime, etc). If cost does matter to you, you probably will not be able to live alone, unless you get a studio in queens for 1k and spend a half hour riding the subway each way.
So if you can't get student housing, but still want to live in manhattan, look for roommates. If you think having roommates will be a distraction, then live in Astoria or Long Island City. But if you find yourself claustrophobic in a packed subway car, and unnerving, I don't know what you do. That's NYC. Use it as motivation, because there will be students living in luxury apartments that daddy bought for them just a block from campus.
FWIW, luxury apartments are at least 2 1/2 blocks from campus. If you want to live a block from campus (or better yet, across the street from the law school), you'll have to settle for spacious, newly-renovated, University-subsidized housing.
EDIT: Also, on a related but tangential matter, all law students are guaranteed
on-campus housing for all three years.