My cat bit me, could I get infected by a rabies if my cat didn't have shots?

Depends. Did you provoke the cat? If so, and the area of your town/city has no known rabid animals over the past year, then it's unlikely. However, get the cat to the vet for his vaccine; the cat may need to be quarantined for a few days, to ensure it doesn't have rabies- though inconvenient, it is more humane than cat euthanasia followed by brain biopsies to check for rabies.