Sheffield has a massive dearth of decent late venues; practically none have open regularly after 4am, which is terribly provincial.
DQ (bland bar/club type place with a seperate downstairs and upstairs, slightly wankerish bouncers) on Fitzwilliam St is open til 4 most nights, with varying music and entrance prices. They soemtimes do afterparties til 8. The Harley on Glossop Rd, by Uni tram stop also has a 4am licence, but is more of a pub which does music and the door prices are either nothing or cheap. The Red House on Solly St has varying opening times, sometimes quite late (3 onwards) and the music tends to be better, with a mix of gigs/parties/club nights etc. Club Shhh, on the Wicker, runs til 8am but only at the weekends and I don't know how often that is.
Hotels - the aforementioned Harley has rooms, not too exorbitant I think, and there's a super cheap Etap out by Sheffield arena (quite near the tram I think). AFAIK most of the city centre hotels are pretty expensive - have a Google tho, there might be some deals knocking about.
Whatever you do, DON'T stay at the Hen & Chickens by Castle Market.
Let me know what night the gig is and I can check what's on.