No, it wouldn't, unfortunately, because being able to move ONTO a tile from a specific side means you can also move OFF the tile from the same side, even though the arrow is going to be pointing in a different direction.
As an example, imagine you have an arrow pointing up, and the tile above it has an arrow pointing left. So you move up then left. The left-pointing arrow tile would need to be passable from the bottom, to allow you to move onto it from the up-pointing arrow, and passable on the left, so you can move off it in the correct direction. However, because the left-pointing arrow tile is also passable from the bottom, you could move down as well.
What you need is a plugin that will let you enter a tile from any direction, but only leave it from the one direction. But you'd then need to give the player an option to reset or go back to the start, that is not linked to them activating any particular object on the map itself, because they might get stuck with no way to get to the object.
If you're happy for the player to be able to backtrack though, the above approach would work.