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That looks like a matte black vinyl dye (paint with flex agent added). I would suggest that you clean the surfaces well with detergent/water, use ScotchBrite to scuff up the surface a bit (and remove any stuck-on crud), then wipe on a coat or two of polymer protectant (Nu-Vinyl, Formula 303, Meguiar's Protectant for plastics).
If that doesn't get the job done (and I think it will), go for the paint. Just be aware that all of that type glass/plastic came un-painted from the factory.
Restoring plastic parts
