If I get in a helicopter and go straight up and just hover there for 12 hours will I come down on the other side of the world?

Yes, this could work. If you were right next to the North Pole. You go up and hover for twelve hours, then land on the other side of the North Pole.

If you're any notable distance from the pole, two problems arise. First, when you're sitting on the ground you're moving with the Earth's rotation at hundreds of miles per hour. Just leaving the ground won't change that, so you'll have to start off with a good sideways thrust so that while hovering you're moving at hundreds of miles per hour relative to the ground. (I mean to say, the ground is moving at hundreds of miles per hour relative to you.) The second problem is the wind that is now moving hundreds of miles per hour relative to you and is trying to push you in the direction of the Earth's rotation. You'll have to fight the wind to hover in place.

Neither of these should be an insuperable obstacle to a sufficiently kick-ass helicopter. But the trip might appear deceptively similar to a regular trip flying around the world at hundreds of miles per hour in your kick-ass helicopter, which I find takes some of the coolness out of it.