I have never seen a bicycle-specific insurance policy. You are probably better off saving a set amount per month so that you have the cash cushion to take care of problems.

A renter's insurance is a good idea to consider no matter what. The deductable of a reasonably priced policy would probably make it cost ineffective to reimburse your expenses (if your bike was < $1500 or so). But the key benefit of renter's insurance is that it should cover your liability if you cause damage to a car (of course, check with an agent before buying to verify that this is true where you live). You'd be surprised how much it can cost to fix a scratch or a dent.