How to subscribe to a mobile game

Apple Arcade the forum you posted in, does not have subscriptions for specific games within in. Apple Arcade is its own subscription that gives youn access to hundreds of full games with no further charges beyond the Apple Arcade subscription.

If you mean a subscription for just a random game in the App Store, then it's not related to Apple Arcade, but usually the subscriptions can be activated from within the game itself through its Settings. Specific steps will vary depending on the game you are referring to.

What game are you looking to subscribe to exactly?