In the Cinnamon Menu, look for Startup Applications. When the startup applications dialog comes up, click on add and where it says command, click on browse and look for the script in the popup file browser. Then give the new startup application a name, and an optional comment and click on add. This should make the script startup automatically when Mint starts.
However, make sure that the script is an executable, by running this in the terminal:
obviously replacing <path-to-file> with the actual path to the file.
Hopefully that helped