If you're planning to leave the rack on the car for months on end, a new rack is going to start rusting as soon as you put it on your car.

If this is for a limited time, say 3-6 months, I'd suggest attacking the old rack with a screwdriver or ice pick to see how deep the rust is (does it go all the way through one or more structural members?). If it's solid now, I'd leave it on until you leave that city, and only buy a new rack then. You can borrow a drill and buy a new wire brush if you want, clean it up, and hit it with Rustoleum or similar to keep the rust from progressing during that short time if you want. But I don't see any sense in giving rust a head start on a new rack.