Is there a way to get plastic to shine again?

There are lots of plastic polishes at auto parts stores. Toothpaste works good too.

Maguires makes some plastic polish that my Dad used to clear up the headlights on his truck. I couldn't believe how well it works. It took a little work, but it worked really well, and made a noticeable difference in the brightness of the headlight.

yes , you can get it to shine again but it all depends on how bad the piece is to determine what you will need. You might be able to get away with just hand polishing or you might have to take a buffing wheel to it.

just used the 3m kit from wally world and did my headlights, they were bad, look like new now.
its a great kit for polishing