Unless you have a compromised immune system (in which case you are gonna die if you go to work anyway) then you can afford to touch the tap. No one is sticking their butt in it so the water itself is clean. Literally just wash your hands with soap and shut it off with your elbow or whatever. You are more likely to get e coli when touching the door handle to exit the bathroom.
As for keeping the cup clean, if they use a dishwasher it will be clean, if they don't you are right to worry, so take your cup home or something.