Use the 'A' key to jump. Some gaps will require that you use the arrow key in its direction as well to jump.
You don't have to restart. If you fall down the first jump, Walk over to the left, up the stairs. There is a rock-roof tile in between two ground tiles. Press forwards against it and press 'A' key.
If you fall down the second hole, walk up to the edge of the short cliff, into the middle, where the ground is partially dug into. Then press down and 'A'.
Yeah, the shortcut is that hole in the wall, next to the elevator. Just stand in front of it, with the upper arrow key pressed, and then press 'A'. Your character will jump up into the hole.
Hope that helps. Sorry for the confusion.