I do not know any way on cultured marble to remove it. You can sand cultured marble, but the finish is a very thin layer. Most of the time you go right to the uncolored base material. In addition, you would have to refinish it with an acrilic coating like they use in the factory. It would look just as bad.
I would suggest living with it, till you can afford to replace it. Just watch for sales at Home Depot or your local hardware store.