The automotive alternator is an AC machine with diode rectifiers, so it will not run as a motor. The old fashion automotive generator, on the other hand, will run as a motor. All you would have to do is connect the battery, the motor starts and begins charging the battery. You might use a flexable coupling, available at the hardware store, or belt drive by putting a pully on the weed whacker. You would need to experiment with pully ratios.
The riding lawnmower does have a battery charging generator, but it won't run as a motor and has low power output.