I was disappointed by how quickly my cultured marble countertop lost its shine, so one day I took out my auto polisher and went at it. What a difference.
Cultured tops are made of polyester resin with a clear topcoat (called the gel coat). This is the same material that's used with fiberglass and it buffs out beautifully. Removing the faucet makes it easier to buff the top.
First wet-sand with 1,000-grit paper (available at auto parts stores) to remove small, shallow scratches and surface stains. Don't try to remove deep scratches or deep stains. You'll risk sanding right through the gel coat. Rinse and dry the top.
Next, buff with Meguiar's Medium-Cut Cleaner or similar polishing compound (also available at auto parts stores). The buffing can be done by hand, but a small power buffer is much easier. Finally, protect the restored finish with a coat of cultured marble polish (available at home centers and bath specialty stores). Follow the label directions.