I'm not sure how you want it, but you could run something like this
on 12v, put a momentary switch some place on the cart, then a relay to the stereo. That way you could push the switch and get an hour of play time on the stereo and it would shut off, and never turn the cart on, you would have to reset it if you listened for over an hour. If you did not activate the timer if would work like it is now only with key. But anyone could hit the switch and make the radio work for an hour, if they can find the switch lol. Or you could use a key operated switch to activate it if you needed.
You could put a switch to run just the stereo, if you mean you forget to turn it off then put some LED lights on the cart that run with it maybe.