Nuremberg has ample bike lanes along busy roads and the Altstadt is pretty bike friendly. For bike hire, try the excellent Ride on a Rainbow.
The best transport around the Altstadt is at the end of your legs. Timed tickets on the VGN bus, tram and U-Bahn/S-Bahn networks cost from €1.30. A day pass costs €8.10. Passes bought on Saturday are valid all weekend.