Here are the steps to defeat Hermes for the badge in Sheep Happens.
First you need to run like 2000-2200m without using resurrection and then you will see Hermes.
Once you caught Hermes there will be a daemon sheep that appears on the right side of the screen. You need to tap on the sheep at just the right moment to defend against arrows.
You can also check how well the battle is going by looking at the life bar at the top of the screen (each time you tap on the sheep, two teeth goes out).
You need to tap 15 times to win. Upgrades let you win with less number of hits. That is pretty much it and good luck! Don’t let the sheep happen.