The Youth Hostel Association runs a hostel in the harbourside area of the city centre:
http://www.yha.org.uk/find-accommodation/south-west-england/hostels/Bristol/index.aspx

It depends on what you call cheap and what kind of arrangements you are willing to accept.

There is a hostel that I have stayed at many-a-times (and will be staying there for one night next week in fact), that is £15/night for a 10-bed, all female dorm. Most of the people who stay at this hostel LIVE there… and I don’t mean they are homeless… they work in Bristol, mostly doing contract work, short-term work, and people traveling on working holiday. As a result, people are much more respectful, clean, and quiet. I would highly recommend it:

http://www.bristolbackpackers.co.uk/

If you can’t deal with hostels, there is a B&B that I use to work at. It’s a 20 minute walk from city centre… but it’s very nice, clean, and you get free breakfast. They have a very small room that they use to only charge £28/night for. You do not have to book in advance, but I would suggest that you do, as more often than not, they are completely full.

http://www.arches-hotel.co.uk/

Hope that helps!