@hesspoint You can shut down the operating system BUT you can’t completely shutdown the power to it. The only way to actually have the pi completely off is to unplug it. By running a shutdown command the led on the pi will still be drawing power and showing you it “isn’t running”.
If you just want to shutdown the operating system/mirror etc and don’t care that the pi still has power… Then, you can do it via the operating system. There is a thing called a cron job which you setup on the crontab. You can set it to run every night at the same time.
Here is a link on the raspberry pi forum about retropi (a different project) that will tell you how setup your cron .
https://www.raspberrypi-spy.co.uk/2018/03/raspberry-pi-retropie-shutdown-button/
There are some MagicMirror modules that might help you do the same thing via a web url that points to the pi at your house. You would need a fixed ip address at your house, setup your router to allow the outside world to talk directly to your pi, and setup the module too. I haven’t used one of them.
I think there is an Internet of Things IOT module that could probably work via IFTTT too.