You would have to experiment with this, but theoretically it sounds like it might work.
Write a VB program that sits in the System Tray and starts when the machine starts. When Windows shuts down (Normal shutdown - if someone hits the power button or the machine crashes, you are screwed) it will close running programs. You could place code in the Query_Unload event of your VB program that does your folder copy.
VB Code:
Private Sub Form_QueryUnload(Cancel As Integer, UnloadMode As Integer)
If UnloadMode = vbAppWindows Then
'windows is shutting down
'your folder copy code goes here
End If
End Sub
Try it and let us know.