In the past, the mummy is right after the front door of the inn. Have Hoagie talk to the mummy. It doesn't matter what you choose, and you can end the dialogue right away.
In the present, go through the front door. Make sure you talk to the mummy before you use it as a decoy for Dr. Fred.
And in the future you need Laverne to have her tentacle costume to get to the first floor. In the second room with a lot of 70's stuff is the mummy. As soon as you talk to the mummy, the achievement unlocks.