Cups are just as comfortable as tampons, once you get the hang of them. (Remember middle school? See, Tampax weren't always your BFF.)
You can maintain cups in a public restroom, but you probably won't have to because you can keep it in for so long. Just wash your hands and wet a paper towel to bring into the stall with you. Then you can safely empty your cup into the toilet, wipe it out with the towel and pop it back in for another dozen hours.
Does the idea of collecting your flow in a cup gross you out? Then likely collecting it in a cotton vagina plug or slab of adult diaper does, too. In that case, good luck.