I second Jeff - you can keep your place neat as a pin, but if your neighbor's a pig you'll have to deal with 'em too.
I don't know if the landlord is required
to exterminate, but if he's/they're worth his salt, they'll be willing to pay for a round. Then you'll have to follow up w/roach baits to make sure they stay gone. Talk to the landlord and see what they're willing to do to clear up the situation. since you're a new tenant I would think they'd want you to have a good impression and be very helpful!
Good luck. I hate those creepy crawlies, myself.