A Guest House is basically a facility which rents out a furnished room on a monthly basis. It's not a B&B because there's no breakfast and because these facilities can house up to 20+ units and it's not an apartment because the bathroom, living room and kitchen are shared. I guess the closest thing we have here is the upscale youth hostel with a private room.
These places start at around 50 000 yen per month but they may have a down payment or shikikin and they may have a minimum stay depending on the place. Here's a link to a selection of guest houses :
http://www.share-style.jp/