You might find you don't need any pets at all when nature is already providing plenty for you. Well, nature and the fast food restaurant downstairs that uses your walled-in "courtyard" as a supplemental dumpster. Just like you'll have to choose whether you want to live Downtown or Uptown, West Side or East Side, you'll also have to decide if you want a roach apartment or a mouse apartment. If you really strike out on Craigslist you might end up with both, but typically one species has already devoured the other in the Darwinian struggle for your Hot Pocket crumbs.
But these vermin only hide indoors because the streets have already been claimed by the rats. New York rats are products of their hometown—they're entitled and strictly business. Dustin Hoffman didn't improvise the line "Hey, I'm walkin' here!" in Midnight Cowboy, he borrowed it from a rat. If you come across some on your walk home, and you will, just keep your head down and don't make eye contact and you should be alright.
And if there's a sanitation strike, it might just be best to stock up on canned goods and stay indoors for a while. On the plus side, you don't have to worry about alligators in the sewers. That's just a rumor spread around so the rats won't seem as abnormally large when you first see them.