Most garage door openers operate by plugging into an outlet. If that is the case, plug yours into an extension cord connected to an outlet that works.
You could have a loose connection at the last receptacle on the circuit that works. If there are any GFCI receptacles on the circuit, test and reset them.
If you can't get a GFCI receptacle to work, try connecting the extended part of the circuit directly, bypassing the GFCI part of the receptacle.
If you plug an incandescent light into the receptacle that doesn't work but shows voltage, does it still show voltage when the light is plugged in and turned on? If you measure voltage when the light is off, but none when the light is on, you probably have a bad connection in the hot or the neutral wires.