You are pretty close with your design idea.
It needs 2 transistors. A P-channel MOSFET that acts a "high side" power switch.
Its gate is pulled high to turn it off.
When the DS3231 Interrupt line goes low, that turns the P-channel on.
The Arduino then drives a simple NPN on to hold the gate low and does whatever the arduino does. When done, it stops driving the NPN, letting the pullup resistor pull the gate high and turn off again.
The NPN is needed so that the pullup resistor voltage does not power the Arduino thru an IO pin's protection diode.
Need to us a Low Rds, Logic Level P-channel FET. A thru hole part could be:
