I assume you swapped out the 3 spd column with an "auto" column because you went with an auto trans? Are you sure it is a reverse switch? Is it possible that it is a "neutral" switch?
Assuming it is a reverse switch and that it is functional (i.e. the switch closes when you are in reverse) then wiring is relatively simple. I'd get power either from an ignition or continuous source, run it through a fuse (5A), then to the reverse switch, and from there, to your reverse lights which need to be grounded properly. When the switch is closed, the circuit is complete and your back-up lights illuminate.
