How do I get into Gee's arena in Hayter's Folly?

This area will be available once you accept the unsurprisingly-named mission Master Gee the Invincible.

From the Borderlands Wiki (to avoid re-typing the exact same information):

He can only be found upon completing the DLC's main mission line, and is available for challenge after the person hosting the game has beaten the Hyperius the Invincible mission. Master Gee's mission is appropriately named "Master Gee the Invincible", and is given by Shade.

If you want to enter this arena, you must have received the mission first, otherwise the lever won't be activated.

This mission is only available once you've completed the main mission of the DLC (source).