The above sounds reasonable to get the power turned on but something has to latch it as the average doorbell press would not be long enough for the Arduino to boot and do it's job.
I would wire the door bell to one pole of a latching relay and have the Arduino trigger the other pole when it's finished doing whatever it does. Get a relay with a coil voltage suitable for the existing wiring and the Arduino can use a transistor to handle the other coil.
Rob