If you really want to find cheap, maybe try searching for Sapporo guesthouses? They can be cheap, but you might have to share the room with other people. I did a quick search for Sapporo and Sakura House and this site came up:
Backpackers Hostel INO'S PLACE -Sapporo Hokkaido Japan- | Cheap & Clean guesthouse Budget Accommodation
It's a Sapporo hostel for backpackers. It says they have long term rates starting at 2,800 Yen/night. Private rooms look like they are 4,200 yen/night. I don't know anything about the quality of the place, sorry!