How to fix this sink
I've used HOMAX Tough as Tile for years. It's an epoxy coating that will stick to just about any clean surface. You can pick this kit up for about $25. It has everything you need for the job (Tri-Sodium Phosphate, Brush, Gloves and Paint). Follow the directions and especially the cleaning section in order to remove soap scum, etc. The process requires 2 coats and will take about 2-3 days to dry but you will be amazed at the end result which will be a candy shell appearance like porcelain. For optimum results, remove the drain for the work and then re-install it when the job is finished. The fumes will need to be addressed by ventilation and the odor is also a consideration.

Note: This kit comes in almond or white. It also comes with a spray can option which I don't recommend. The spray gets all over everything and takes a long time to dry.