Exists a means to power up a Mac Mini from another location?
As you define it, no.
You could, nonetheless, get an additional computer system or network-controlled tool with the capacity to reduce off/restore power to a tool, and also from another location run that (I'll search for such a controller and also link it below soon ). After that simply set your mac-mini to auto-boot after a power failing.
Modify : instance- http://www.42u.com/dataprobe_iboot_remote_reboot.htm
Depending on just how your Mac Mini is attached to the net (wifi, ethernet, firewall program, age of the equipment, NAT, ... ) it needs to be feasible to do it making use of Wake On LAN.
You could, as an example, make use of a GUI software program like WakeOnLan.