Unfortunately, things like this were never meant to last and they age terribly. Art museums are actually struggling with preserving art pieces made of plastic, in fact! So I'm going to turn to my art background and suggest that you get acid-free board (such as book binding boards, just make sure it specifies acid-free) and acid-free book binding glue. Cut the boards to the same size as the sign and glue them together at a right angle (making the same shape as the sign - you're making a strong backing for it). Once it's dried, glue the sign to the backing. I usually hesitate to permanently glue things like this, but I think it badly needs the structural support, as it looks very fragile.
You will also need to keep it out of the light as much as possible. Sunlight is particularly bad, but from what I understand, even artificial light can be damaging to plastic. Keeping it away from sunlight should be a priority, anyway. Good luck!!