Plastics can be polished like a car paint really, it's just a little bit softer. As long as the color is on the inside, as Mike mentioned, you can polish it. If it was me, I'd clean the exterior with wax and grease remover, which is a very mild solvent, that removes contaminates like tar, grease, etc. It's available at auto paint stores. Then I would try using an auto buffer to the outside and see how it shines up.
I did a Pabst sign recently that was yellowed, I sanded the exterior with 1000 grit sandpaper, removing the yellow, and then buffed it. It doesn't usually give you a super high shine, more like a semi-gloss or better when done. The sign in the picture I just buffed out, which improved the shine, and didn't need any sanding at all.
There was one plastic sign that I like'd better flat, so I just sanded it and left it.
You going to build a can for it?