The game runs on Java, so you'll need to install it through Mint's software centre. It's probably called something like openjdk or sun-java-jdk.
Once you've done that, go to the folder you extracted from the Domination download, and right-click on the run.sh file. There should be a permissions tab, from where you can tick a box to make it executable.
Now you should be able to click on run.sh and it will launch Domination.
Hth, I don't have Mint installed so can't double check those steps! Let us know if you have any more trouble, and if I've made any mistakes hopefully somebody can correct me!