[Firebird]

Search for:
youth hostel
hotel
ryokan (traditionell hotel)
minshuku (kind of like a B&B, mostly available in small to midsize town)

If you want to move around alot, why would you want to rent a place for the whole month? Japan is quite big, not everywhere is a daytrip away!

I guess your best option comparing price/comfort would be a minshuku, but most of them don´t speak alot of english and are not in all the guidebooks, try the local touristoffices or inside the trainstation on arrival.

You could also look into "gaijin houses" but im not sure if they rent if your on a tourist visa.