I would like to travel to Munich for a month and am considering accommodation. I realize that hostels are very cheap, but I would be willing to pay more for a private room. Renting an apartment for a month is probably too short. Do hotels usually offer discounts for a month-long stay if I contact them? If so, around how much might it cost? Also, are services like Airbnb a good option?
I'd give Airbnb a shot, though I never tried it personally.
In Germany such short-term apartment rentals are indeed nearly impossible. However you can maybe find something called WG - a roommate arrangement. Some of them may accept a short-term rent, maybe to fill for a sport which would be otherwise empty. A popular website to search for such places is http://www.wg-gesucht.de/.
If you are young, love company and can turn an eye to a thing or two, maybe you could give a shot to https://www.couchsurfing.com/ too. That maybe absolutely free anyway...