Melamine kitchen cabinets

http://www.wikihow.com/Paint-Melamine

well they probably didn't sand them first you will need to sand that new layer off and put an enamel paint back on

See if the landlord can help you out with perhaps supplies and/or help. It should be them that repair what a previous tenant did. Make sure to ok anything you do with them so they don't make you undo it or charge you for changes to the property.

Don't do anything until you talk to your landlord. Once you can legally work on them, get it in writing, you can sand everything off.

IF they are covered in plastic that comes off when you sand, then you can peel the rest of the plastic off. The wood underneath is a compressed wood, so you will need to seal it very well with primer under the sink area, so water can never get into the "wood."

Use an oil primer.

Then if you can, buy PPG/Porter Paint's Advantage 900, and paint them with that. Semi Gloss sheen. It is a water based paint, but it is VERY strong. Let each coat dry overnight before proceeding. No need for a clear coat sealer at the end.

I would like to second the importance of not getting the "wood"/ compressed wood stuff wet!! It is the cheapest most awful sawdust-ish mess and that's when it is dry!! It will totally disintegrate if really soaked. Do try to seal it on any edge / every edge.

Sanding is probably a good place to start for this project.