It would have to be a running process. If it is running, it could be that windows has hidden it. First right click on you system tray, then click properties, then the customize button. Now find the outlook icon and select always show from the drop down menu. Tada!
You have new mail does appear, OUTlook does not neither does Outlook express?
likely reason is because there is no need for it to appear there ,it adds new mail notification ,you could add it to the lower toolbars quick launch ,if you need it.if you don't all ready have it in quick launch ,
right click on the lower border ,go toolbars ,click on quick launch ,right click on the outlook icon in start and you should be able to send to quick launch ,or drag and drop it in quick launch, good luck
System tray [or notifications area] is used mainly for ... yep.. notifications. So new mail events pop an icon there, your AV icon shows you it is doing stuff, a temperature monitor software may report there, some icons there allow configurations. Do as Caper suggests, drop it into the QuickLaunch area. I think that to get an icon into the system tray requires coding?