I don't have Outlook 2003, but older versions have the possibility to set different ways of notification for a new email in the options, like a "you've got mail" pop up window or a sound playing. Do these work?

There's even a remote possibility that the systray icon appears again if you rechoose it in your options after finding that the pop up window works.
If not, stick to the pop-up window until you found the cause. Personally, I very much prefer that nice unobtrusive icon in the systray, but you might find a pop-up message better than nothing at all.

If no notification at all can be made to appear, the problem possibly can be solved by a repair or uninstall/reinstall of Outlook from the Office CD.
Of course, such an action wouldn't harm now, nor would a online malware search with, but I doubt if they would help.

Hope this helps.