Briefly "wake up" Arduino in response to an event
Easy answer: Don't.

Have the Arduino sleep most of the time, and let the door handle invoke an interrupt that wakes it up. Then do whatever processing is required, and then go back to sleep.

This won't work as well with a full Arduino as it would with a bare MCU since the rest of the board has its own quiescent load, but once you get it working with an Arduino you can get a bare MCU, wire it up, write the code to it, and away you go.