22 years old, looking for a room from the 1 of November

Housing is expensive in Germany and with a tight market in most cities, especially difficult for foreigners to find. The cheapest and often easiest approach for young people is usually to get a room in a shared apartment know in Germany as Wohngemeinschaft or WG. And the best possibility for this is to look for pin-boards at Universities or other educational institutes where people will hang advertisements offering or looking for such rooms. Just getting a place for such a specific time period might complicate things but often students go away for a semester and want to sublet their room and thus one might even get a furnished place at an extra cheap price for a limited period of time. 

To rent a place on one's own means finding a place free and having the landlord interested in renting it for a short time. This is not so usual except for furnished places, which can be what you are looking for - but then often much more expensive than a similar unfurnished. One is usually better off getting used furniture or buying simple things at Ikea but since you are only staying 3 or 4 months, it is hard to judge if it pays off. People staying for a year or longer are almost always better off getting their own furniture and selling or giving it away when they leave rather than paying for a furnished place.