I have a few small stains and a spot of dried 'Crazy Glue' on my off-white cultured vanity marbletop and I wanted to faux paint a design over those areas. Can this kind of surface be painted on and if so, what type of paint should I use? I prefer not to use oil based paint.

You can paint this surface but the paint will be softer than the faux marble. You need to scuff the surface with 320 sand paper and wipe it down with mineral spirits. Zinsser 1-2-3 primer should give you a good base for your top coat. I can't recommend a water based paint for this application since there may be standing drops of water on a vanity. Why the opposition to oil?

Thank you for your recommendation! The only reason I don't like oil based paints are because of the fumes and the messy clean up. Could you recommend an oil based paint that would work on this type of surface?

I thin, because of the water, a marine grade epoxy would work best on the vanity. You may need to call around to find this epoxy or just go to a boat supply dealer. Some home centers also sell a paint designed to reglaze a tub, porcelinized paint.