there are lots of b&bs, hotels and hostels in the city centre. did you try one of the booking engines online? i would recommend the Aachen Hotel, cheap but good and very central.
plenty hotels, just avoid match weekends as you will find it hard to get a room. Any other weekend should be ok as the place will be quieter but maybe not this year being capital of culture or something like that
Unfortunately every weekend of the football season is a match weekend. I believe there is a team called Everton who play on the weekends that Liverpool are away .
Try LateRooms.Com. They usually have a good choice. You can even use them to research accommodations as they have photos and reviews.
Premier Travel Inn (city centre) Vernon Street , great city location 5min walk from matthew street, heading there myself for this weekend. Now in fairness the end of the street is a bit run down but its cheap 128stg for two nights (double room) - its on the bus route as well from the airport so this is handy saves taxi fare. Breakfast isnt worth the money but the BHS is only a few mins walk away which is great for breakfast about 3stg for 10 items.
I have found that you need to book weeks if not months in advance due to the football situation. I usually end up staying in Manchester and getting the train to Liverpool.