The immigration clearance is usually very quick in FRA - no more than 10-15 minutes, at worst. But as you are connecting on an international flight, you should not have to clear immigration. Even without lounge access you will probably have enough time for a shower, but it could be tight at that time of the morning. The walk to the C-gates from A and B (where the *A flights dock is very long - you will need at least 15 minutes to get from A to C. Good luck.