Dan, where are you located? I'm about 20 miles west of Pontiac, and have been messing with Sprites for over 35 years. Always willing to help.
Rust in the rear spring boxes is probably the worst thing that can happen to a 1/4 elliptic car like the '63. Fixable, but a major pain in the posterior. Common points for rusting are at the bottom of the A pillars, at the lower door hinge, bottoms of the rear fenders, and the rocker panels. Battery hold down issues often cause the battery to wear two small holes in the firewall. The battery tray itself is another common area for rust, but is easily repairable.
Look at the wiring for obvious "lamp cord" fixes, and general bodging.
Plan on replacing the rubber bushes in the suspension, as Arizona weather is hard on them. (I lived there for 8 years.) Pay attention to the rear trailing arm bushes, as when they go bad the rear end can do some strange things to the handling!
The '63 is a sweet car, as you get the 1098, rib case transmission, disc brakes, and the better handling of the 1/4 elliptic rear springs, but have the extra elbow room of the side curtain doors, as well as the quaintness of the early dash and the pull start. I've got one here, that unfortunately has pretty well used itself up as a sacrifice to the rust gods. It was a California car, but spent the majority of its life living about 150 yards from the Pacific. Salt water and Sprites don't mix!
Jeff