You shouldn’t need to install again - indeed, you don’t even have to have Cubase installed to redeem your Authorisation Code, just the eLicenser Control Center.
Run the eLicenser Control Center, make sure your USB eLicenser is attached if you are upgrading to Cubase Artist or Cubase Pro, then press the “Enter Activation Code” button. Enter the code and follow the process through. You should finish up with a full licence for Cubase at the end of the process, which allows you to use Cubase.