replace the lamp socket, or rebuild it if you prefer with new insulation. replace the wire with a similar type of same or larger gage, and also as it is a metal fitting attach a ground wire to the fixture itself for safety.
Looks pretty straight forward. As far as bonding, they look like they attach to the box with decorative nuts and the studs would create the electrical bonding. It also looks like they have already been retired with brown lamp cord, they may be good to go as is. Check to see if the existing lamp wire has one conductor marked with a couple of small ridges. That is an indicator for getting the polarization correct (not that it,s a big deal on a simple lamp) but the power (black wire) should make contact with the little button and the common (white wire) connects to the threaded shell.