You would have to power the RTC separately. ATXRaspi is powered directly from uUSB so it has power all the time, waiting for a button press event. You can simulate a button press from firmware, but that will be a little complicated. Your RTC would have to have the intelligence to send out a negative/GND pulse to ATXRaspi to wake it up. So I would not even try that if I were you. Maybe a custom ATXRaspi version that has an RTC could do it, but that would require a more complex controller (different chip, with extra firmware). I suspect you also want to be able to change the schedule so it gets complicated fast.
ATXRaspi is really designed to be manually operated where a SSH login is not available.
You can probably find a way to use wake-on-LAN on the Pi...