I can make a few suggestions when it comes to places to stay in Cluj. 

The Retro Youth Hostel

Str. Potaissa 13

This is one of the best hostels I have ever stayed at.  It is situated right next to an old medieval wall, the staff is helpful, and it is very clean.  You can either stay in a dorm room for about nine dollars a night, or stay in one of the two single rooms they have for a little more.  They also organize frequent day trips to the outskirts of Cluj and beyond.  I could go on praising them, but instead I will direct you to their very slick and up-to-date website here.

4 U Hostel

Str. Cetatii 26A

This place is on top of a hill that overlooks the city of Cluj.  The bathrooms are not private, and most of the rooms are doubles.  So if you are coming with a friend you can get a room for about eighteen dollars each per night.  A free breakfast is served.  I would use this as an alternative if the Retro Hostel is full, which is frequently the case during large events in the city.

Hotel Sport

Str. Cosbuc 15

Hotel Sport is located inside the large Parc Central.  It is hard to explain the feel of a place that was built during Communist times compared to newer structures.  Lots of concrete, and the staff may not speak English but will probably speak French.  The woman behind the desk is likely to be sporting a hint of a moustache and wearing a disapproving smile.  I have stayed in several hotels like this throughout Romania, the most depressing of which was in a place called Faget.  Hotel Sport is not as cheap as the two hostels, but you get your own bathroom which is a plus.  In fairness the rooms are clean if worn down a bit.  You will pay about sixty-five dollars a night for a room with a double bed and a television.  A warning to those who like to sleep in late though: This place is located very close to a soccer field. On the weekends you will be greeted by the sound of cheers, jeers, drums, and kazoos wafting through your window in the late morning.