I dislike bleach for cleaning items that contain food unless absolutely necessary, so I decided to try my favorite cleaning method - baking soda and distilled white vinegar. I got the cup wet, then dusted the inside with the baking soda. I put the cup in the sink and poured in some vinegar, enough to make the baking soda fizz and turn it into a paste. Using a clean cloth (I don't use paper towels, but they would also work), I rubbed the paste all around the inside of the cup and the stains came right off. I finished by washing it the normal way and letting it dry.
So far this has worked for both porcelain and bone china and hasn't had any effect on the hand-painted or gold leaf exteriors.