Is there ANY way to Power Up a Mac Mini remotely? Just to be clear, I don´t mean wake up from sleep but boot up.
As a workaround you can set, in the energy settings a scheduled time for the remote computer to start. At least if others have powered down the computer you know it will power up everyday on its own i.e. 6:00AM I am setting up a remote MAC in the Philippines and others may power down instead of log off so I know if for example I set it to start everyday at 9:00PM philippine time it will be acccessable to me in the US everyday at 6:00AM.
Another idea if your in the US is to use iDevice, a switch that you plug into the power outlet then plug the MAC into. These are great devices that use apple home kit and I use them to power cycle my remote cameras that freez once in awhile and the company who makes the camera said to power cycle to restore the cameras. These devices use their own internet access and work great unfortunately they only make the 110 vac version. Also you must have the MAC set up to restart on power failure.
Hope this helps
As you describe it, no.
You could, however, get another computer or network-controlled device with the ability to cut off/restore power to a device, and remotely operate that (I'll try to find such a controller and link it here in a minute). Then just set your mac-mini to auto-boot after a power failure.
Edit: example- http://www.42u.com/dataprobe_iboot_remote_reboot.htm