Those who get the message that you should use PC, it means that you have conflicting packages and application manager is trying to be nice and not force-remove them.

This can be caused by an exact dependency from an installed package to one of the system packages. It seems like qt4-phonon, qt4-webkit are good candidates for causing this. If you have e.g. midori installed, that would be a good candidate for removal before proceeding with SSU.

You should probably remove potential blockers (anything from devel is a good candidate) and try again. You can also use apt-get which may or may not work (and you need to reboot at the end yourself).

Before resorting to apt, you need to move apt caches to some larger medium than rootfs. You can do this with:
mv /var/cache/apt /home/user/.apt-cache
ln -s /home/user/.apt-cache /var/cache/apt