Not good idea . Classic mononucleosis is caused by the epstein-barr virus (ebv). Other viruses can cause mono like diseases. Once you've had ebv, you get life long immunity. So you won't catch it again. But it's unwise to share glasses or other materials you put in your mouth with someone who's sick. Btw, ebv can be excreted in the urine for months after a person has recovered from the acute illness.
No. Generally, mono (epstein barr virus) is a once in a life time infection for the vast majority of people. If you had mono, then you have antibodies against the ebv. If you are on immune suppressing medications or have other chronic illnesses there may be a chance you'll get infected again. But if you're in the latter category, you shouldn't be sharing drinking containers with others - yuk.