>update a program like Adobe Acrobat Reader, and save it for the next re-boot

Sorry, no help available from me on this point (I'm not running Knoppix).
From what I hear it can be done, so perhaps someone else can take this up.

>install software not covered by the apt-get utility?

Basically, you try hard to avoid this
OK. First thing that comes to mind is "klik", there is a separate forum for
this on this site a little below "General Support".

Then there is the "classical" way of obtaining a "tar"-file (that's TapeARchive
and does not have to involve a tape) which you have to unpack, and then
use the commandline: "./configure", "make", "make install". Instead of the
last command you better use "checkinstall" because that builds and
installs a "*.deb", so that the apt-get utility will get to know about this
software e.g. for a later deinstallation. But all that would take place on a
system installed to harddisk, not a live CD.