First wash your hands at the sink- as you would when changing a tampon.

You can then take a small bottle of water into the bathroom stall with you.  Remove the MCUK, tip the contents into the toilet bowl, rinse with a small amount of water and re-insert. Or you can simply tip the contents into the toilet bowl, wipe the MCUK with toilet paper, and re-insert.  You can give it a more thorough clean when you get home.

Remember that you may not need to empty your menstrual cup as often as you would change tampons or pads, because the MCUK holds more.