Finding a place to sleep isn't difficult, as you will probably be approached by people offering accommodation. Outside of the tourist season you can probably bargain pretty hard on the prices.
The Qufu International Youth Hostel is a very nice place to stay. With a friendly staff, and entirely new building this is an excellent place to sleep while exploring Qufu and its surroundings. The hostel is located at the northwestern end of the main street inside the city walls. It is ¥140 for a double/triple with private shower.
