Apple - Is there a way to power up a Mac Mini remotely?

Depending on how your Mac Mini is connected to the internet (wifi, ethernet, firewall, age of the machine, NAT, …) it should be possible to do it using Wake On LAN.

You could, for example, use a GUI software like WakeOnLan.

Not sure what you're trying to accomplish, but it's also possible to use the Energy Saver preference pane to schedule the machine to startup at a set time.