I run the Kodi media center on a Raspberry Pi 3 in my living room. The Pi is simultaneously used as a sort of microserver, so to keep the CPU load low, Kodi should only be run when it is actively used.
The remote app Kore includes a Wakeup-Button that sends a Wake-on-LAN packet to the Pi. Normally it is used to wake up the computer from a sleep mode, but this project utilizes the WoL-packet to start Kodi.
Put kodi-starter.sh and wolscript.py in the /home/pi/Scripts directory.
Copy wolkodi.service to /etc/systemd/system.
Then you can simply run
sudo systemctl enable wolkodi.service
sudo systemctl start wolkodi.service
to enable Kodi-execution on WoL-packet.
This script file could also be used for similar projects. It listens for the WoL-magic-packet and returns with exit code 0 upon reception.