The pop-up notifications for Hot Mail come from Windows Live Messenger. Often times Messenger becomes deactivated (for whatever reason), and you have to reopen it. And/or you might not be signed to Messenger in at any given time -- so you need to check that too, and if you're not signed in to Messenger, then do so. Your pop-ups will immediately appear.
If your Messenger is in need of repair, close Windows Live Messenger completely, then open Programs and Features in the Control Panel (icons view), click on Windows Live Essentials 2011, then select the ‘Repair all Windows Live Programs’ option. This will hopefully repair it for you and allow it to work properly afterwards.