After the mailman has opened and closed the lid, and the message has been sent, then the circuit will wait for me top open the part with the lock. It will then reset the circuit
You see how important context is. Instead of asking about how you think you should solve the problem, telling us what the problem actually is will allow us to propose a much better solution.
The circuit you need is called a flip flop, it has two states and can be flipped into each state with a pulse.
The first pulse you can supply with your push button which will then latch the circuit on and power the Arduino until you send a second pulse, from your Arduino to turn it off.
You can even do this with a thing called a latching relay which operates in the same manner. The best type of latching relay for this is the two coil type, pulsing briefly one coil turns the relay on and pulsing the other turns it off. That way there is zero current drain when it is off.