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I can understand your concern. Do not worry, and I will clear your doubt. Infectious mononucleosis usually caused by EBV (Epstein-Barr), and it can spread through saliva mainly. So sharing of the objects like water bottle or glass object can spread infection. So you are at risk of spending it. But do not be panic. The symptoms are fever and sore throat, mainly with enlarged cervical nodes. Splenomegaly also can occur. If fever, malaise, and sore throat occur than investigate by CBC (complete blood count) and serological test for EBV, which includes ESP (extrasensory perception), Ig M (immunoglobulin M), and Elisa (Enzyme-linked immunosorbent assay).
Infectious mononucleosis usually resolves by its own with time. Anyway if you suffer from fever and sore throat that will discuss further and you can consult a nearby physician for examination at that time. It is advisable not to share an object with an infected person. I hope this will help you. Get back with more query if you still have doubt.