There's a second hand bike shop up beside the English language "Cinema". I think the street is called Nymphenbergerstr. I picked up a good quality bike for €100 there. The bike I bought had been sold to someone else at the begining of the summer and they bought it back off him at the end of the summer. I then picked it up as a cheap winter bike. I'm sure you could come to a similar deal with them. The bikes in the Flohmaerkte generally require some extra work on them, so you'd need to be able to estimate how much this will cost to do yourself, or get done at a shop.
I'm sure a month's rental will cost you more than €100. In fact, you can get new bikes for about €200 and often less. I think there were some in
OBI for €179 the other day, and once spring comes you'll find some cheap ones in many of the supermarkets. Of course the quality of these isn't great. You need to ensure all nuts and bolts are tight before cycling off, and then check them again after a few weeks. €300 will get you a semi decent one in
Karstadt or Radlbauer or any of the lower end bike shops. And you could try to sell that for €150 or €200 at the end of the month.