Sadly (for me) the Scroll bar on Ubuntu 12.10 are still the same as Ubuntu 11.10. I particularly don't like it. So I tried the same method to disable the overlay scroll bar on Ubuntu 11.10 and Ubuntu 11.04 and Ubuntu 12.04 and it works! (I just tested the disable method.)
# echo export LIBOVERLAY_SCROLLBAR=0 > /etc/X11/Xsession.d/80overlayscrollbars
If you try the other methods, please let us know if it works!
PS: The scrollbar arrow is not showing, maybe a little bug.