Use a vibration sensor? PS, the answer is as good as the question?
How does the sensor sense vibration if it is off?
Agreed, whatever circuit turns on the Arduino needs to act without it since it would be off.
It's a good simple question. Re-read it.
Any external circuit to do so will take more energy then putting the Arduino to deep sleep and let it waking up to "vibration" for example with a vibration sensor.