More than likely you would not need any local anesthesia. If you did it would probably be some slight infiltration for the gum tissue.
I've never had anesthetic or anything to put on a permanent crown. It's not a painful process, particularly if a root canal is done and healed before the procedure (which it sounds like you're well on your way.)
The tooth is already completely prepped under the temp crown. He will pop it off, clean off all the temporary cement, and pop on the new crown with permanent cement. He will then likely file the new crown here and there to adjust the bite if necessary.
It's not painful at all.