First you have to sand it with 800 grit sandpaper. You can use our silicon carbide sanding discs with a 5" backer:
http://www.defusco.com/800-grit-silicon-carbide-sanding-disc-velcro-backed-p-1693.htmlhttp://www.defusco.com/rigid-backing-pad-velcro-p-1942.htmlhttp://www.defusco.com/flexible-backing-pad-velcro-p-1941.htmlThen you take a muslin wheel and a polishing cake (white for light, black for dark) & buff it until you get the gloss you want.
http://www.defusco.com/buffing-wheel-muslin-p-2362.htmlhttp://www.defusco.com/prep-tools-polishing-cakes-c-89_702.htmlUse the muslin wheel on a polisher or a variable speed grinder at a lower RPM (not to exceed 6,000 RPM). You spin the muslin wheel into the polishing cake to get some of the cake onto the wheel and then apply it to your cultured marble counter.
The sink may not be salvageable if it is in really bad condition. There is a gelcoat on top of the cultured marble, and if it is shot, well, you may have to replace this unit. If it is fixable this info should do the trick. I got it from an expert in that field!