If a watch has a scratched up plastic crystal, chances are very good that it can be cleaned up easily.
Materials: Wet sandpaper in 600 and 400 grit, 320 optional for really bad crystals like this one. Grits are approximate, different brands might not have exactly the same ratings. Padded mouse pad, optionally a soft cloth if the mouse pad doesn't have an appropriate soft cloth texture. Brasso or Wright's Silver polish Water
Water is to keep the sandpaper from clogging with plastic residue. You may want to skip the water if using more sandpaper is less objectionable than getting the watch wet.
Wet the coarsest sandpaper you'll be using and place it over the mouse pad. Check the height of the crystal, the angle you'll need to sand, and make sure you won't sand the bezel--If that's a danger, tape it up or remove the crystal. Sand in one direction, across the direction of the worst scratches by moving the crystal back and forth over the paper. You may decide to leave some traces of very severe scratches, or scratches around the edges if removing them might make the crystal too thin.
Continue with successively finer grades of sandpaper until you get to 600. With each new grade, sand across the direction of the old one, and stop when the old scratches are no longer visible. You'll need to dry the crystal to check progress.
The first time you use it, put a small amount of polish on your damp cloth or mouse pad. If you re-use the same cloth or pad without cleaning it, you may not need fresh polish every time, just re-wet. If using a cloth, place it over the pad. Polish in a circular motion, again concentrating on the edges until the crystal is clear. Again, you'll need to dry it to check progress.
(Note that this case was floating around in my parts, and there is still a visible scratch at 7--That's on the inside. unusual to find an inside scratch on a complete watch, and they are a bear to remove. Not gonna bother here, since it's not going to be used.
Addendum by reader:
An alternate method of sanding crystals is to use sanding sticks of decreasing coarseness followed with a polishing compound. These sticks may be purchased in hobby shops and have the advantage of being flexible. This allows them to conform to the shape of the crystal when polishing. The method used is to place the crystal on a suitable surface (I use a mousepad) face up and sand holding the sanding stick in the thumb and first two fingers of the hand. Using this method there is no danger to the bezel since the crystal has been removed. My results have been quite favorable.