You would be surprised how good the door will look after a good cleaning with a scrub brush. If that door is stainless steel use an automotive wheel cleaner.

Once clean, polishing is an option but the texturing may be a pain to deal with. I bought a bunch of wheels and compounds for my buffer that work great. Check out the Eastwood website for their kits.

Take it apart, then some simple green and the buffing wheel (tumble the coin eject). I believe Marco or Pinball Resource may also have "skins" for them if you just want to go that route. While you're at it have at the ball plunger and legs too. A good five minutes or so and they will be looking like new.

that will scratch the daylights out of it.
try nifti and a toothbrush first

keep in mind its a pinball machine not fine artwork. scrub it, clean it, play it. If you're excited enough to remove the door and strip it down you could soak it in something .. I doubt its worthwhile. If it was here I would put it in the ultrasonic cleaner but thats probably not a valid option for you.