Copper is very unstable. It will tarnish almost immediately, and corrode. Lots of 'decorative' copperware is lacquered to keep it bright, so you wouldn't get any benefit from it anyway. I've heard, though, that if you drink from a copper vessel a few copper ions leach into the water and that's good for you.
Brass CAN contain lead but I'm sure any brass meant for eating off of would be lead free. You can get test strips at the hardware store for testing for lead. It's less reactive than copper, probably cheaper too, but it still tarnishes quickly, and it's still very expensive. Most 'brass' you see in stores etc. is really tin or zinc with a brass coating, because brass is so expensive these days.
I wonder if Nalgene makes cups. They make water bottles that have some anti-microbial action built into the plastic.