The fuel pump in a 1993 Jeep Grand Cherokee turning on when the key is turned to the "on" position (but not started) and then shutting off after about a minute is normal behavior. This is a *fuel pump prime* or *self-test*. The system is checking to ensure the fuel pump is working correctly before the engine is started.
If it *doesn't* prime for that short period, that would be a problem indicating a possible fuel pump, relay, or wiring issue. But the behavior you're describing sounds perfectly typical for that model year.
