You're in an environment where you can't take long (or worse, randomly timed) down times. It's a totally ordinary thing.
Normally, you'd want to upgrade the software in one step, and restart the service that depends on it in a different step.
Here's how.
First, get root on the machine. If you can't, you shouldn't be messing with this anyhow.
Next, do
apt-get update
This ensures that the correct software is installed.
Next, thank Ron DuPlain for making this all so clear.
Create a file /usr/sbin/policy-rc.d with permissions 0755 and contents here.
Next, apt-get upgrade
Now, the software is all upgraded, and you can restart at the time of your choosing.
It turns out that there is a much simpler and less cover-plate-prying-up way, and many thanks to Jehan-Guillaume de Rorthais (ioguix) for pointing it out. First, find out which clusters are involved:
pg_lsclusters
Next, edit each one's /etc/postgresql/[version]/[name]/start.conf so that the uncommented line reads,
manual
If there are a lot of clusters, you might wish to semi-automate with something like this:
$EDITOR $(pg_lsclusters|egrep '^[[:digit:]]'| perl -pe 's!^(\S+)\s+(\S+).*!/etc/postgresql/$1/$2/start.conf!')
Then ensure the uncommented line reads,
manual
do your
apt-get upgrade
Restart at your leisure, and reset the contents to auto as above, if desired.