Here is the full list of all 9 No Time To Explain Remastered achievements. It takes around 5-6 hours to unlock all of the achievements on Windows.
Play through Straightjacket Darkness universe without dying, with a horse head.
Eat every fatcake in fatcake universe with a hamburger for a face.
Collect all the hats.
Beat the scrolling shooter segment without dying, as a dinosaur.
Beat Rejected Green Clone universe in under a minute wearing a monocle and top-hat.
Finish the whole damn game.
Collect half the hats.
Dance to the first boss and don't stop for nothing.
Finish Shotgun universe in under 60 seconds.