I'm not used to proving isomorphisms, so I don't know if the following is sufficient. But I think I can find the mapping: Let $a$ be the permutation that swaps 1 and 2, $b$ be the permutations that swaps 1 and 3, $c$ be the permutations that swaps 1 and 4. Two observations: 1. Every element in $S_4$ can be expressed as a sequence of those three permutations. 2. They each have order 2. So we can try the following map. If you put the cube down on a table facing you, $a$ is a 180-degree rotation on the horizontal plane, so that the top and bottom faces stay but the front and back are switched, and left and right are switched. $b$ is a 180-degree rotation on the vertical plane that's facing you, so that front and back faces stay the same $c$ is a 180-degree rotation on the vertical plane that also dissects "you" in half, so that left and right faces stay the same. Each of those rotations have order 2, and every element in the cube rotation can be expressed as a sequence of those three. Just try performing $abc$ or $acb$, etc, and see what I mean. This is rather hard to describe without showing the actual cube.

I used this in the exam: to the main diagonals of the cube, assign numbers $1$ to $4$. each orientation-preserving symmetry permutes the diagonals, so we can assign to each orientation-preserving symmetry a permutation of numbers $1$ to $4$, and it's easy to see that this is really an isomorphism.