If you are saying that the Outlook System Tray notification doesn't appear unless Outlook is running...



I think that's the way Outlook works. If Outlook is not running, notifications associated with the application cannot display.



On the other hand, if you're saying that Outlook is running, but you're not seeing its notifications, my guess is that the notification icon is hidden:





Click on the leftmost icon, the one that points upwards, three to the left of the circled icon. This will show all hidden icons. Drag the ones you want to see to the main part of the System Tray, and they should stay there.