The bit over the bridge is called Handbridge, where you'll find the Carlton, which used to have a reputation for being the pub that everybody drank in that was barred from the other pubs in Handbidge, though it sems to have changed - still haven't worked out where the old regulars have gone.

Several pubs near the station have had sessions or folk clubs over the years and I wasn't in Chester 20 years ago, so I can't help.