Electrical – How to ground a new outlet that only has the black/white wires

The neutral and ground MUST NOT be bonded at a sub-panel. They should only be bonded at the main service panel. If you bond them anywhere other than the main service, the neutral return current now has multiple paths, including though your ground wire.

You should be able to buy a second bar for the sub-panel if it really is meant to be used as a sub-panel. The neutral bar will need to be isolated (it should have plastic insulator separating it from the case). The ground bar should be bonded to the case.

The ground wire from the house must be connected to the sub-panel ground. Since it is a detached building the ground rods are also required by code and should be tied to the sub-panel ground. The ground to the service is the most important because it will provide the low resistance path back to the service in case of a fault.

Assuming you are in the US, this could be a half-switched outlet. Do you still have the old outlet? Look at the side and see if the metal tab between the screws is broken off on one or both sides. If it is, then (at least) one of those black wires is probably controlled by a wall switch somewhere, resulting in one outlet being on all of the time and the other switched.