According to Redditor TheFotty, there have been a few instances that Evony has swapped different levels on varying servers. This means that you may never experience the Shooting Barrel Game.
However, when you encounter the Shooting Barrel Game in Evony, you have to collect stronger weapons as you shoot the oncoming barrels.
It is up to you to destroy the barrels before they reach your character. You will allegedly be rewarded for every level that you successfully complete.
It is worth remembering that the Shooting Barrel Game, just like the Mysterious Puzzles, has no influence on Evony's main objective. That being said, you will earn rewards for completing levels and solving puzzles.
Several players do not invest time in the Shooting Barrel Game as they can procure the rewards elsewhere in the game.
It is advisable that you only play the Shooting Barrel Game for fun, as it will not help you progress in Evony.