C (gcc), 50 bytes
An explicit carry flag wasn't necessary after restructuring the loop to end as soon as no carry would happen. The 9->A adjustment is performed during the loop check.
f(char*s){for(s+=9;(*s+=(*s-57)?1:8)>90;*s--=48);}
##Original version: 71 57 bytes
This version uses a carry flag to propagate updates: I set it to truthy to begin the increment. The string is modified in-place and only accepts 0-9, A-Z. The tricky part was making sure that 9->A got handled correctly on carries.
Edit: I repurposed the input pointer as the carry flag.
f(s){for(char*t=s+9;s;)*t--+=(s=++*t>90)?-43:7*!(*t-58);}