sorry, only just saw this.
have not much time now, but
Brauhaus Joh. Albrecht at the backside of the Rathaus is nice for a drink and some typical German food (gutbürgerlich = plain fare). In the Rathaus there's
Parlament - not exactly cheap, but very nice, very good food, decent cocktails.
On Börsenbrücke there's
Finnegan's Wake Irish Pub - my favourite pub, also decent food.
Some other ideas, but I haven't got time, google yourself:
Fleetenkieker, Kartoffelkeller,
Cremon (VERY nice, good for Flammkuchen and fish of course).
Touristy: Friesenkeller at Alsterarkaden. Or Deichgraf.