How do I update other Ubuntu partitions without rebooting into them?

There are a couple of techniques, the simplest is to use a chrooted environment. In this environment you can then run sudo aptitude update and most other command line tools.

Alternatively you could create a number of virtual box virtual machines that used the partitions as their hard drives - I have not tried this.

For the chroot environment mount each partition in a directory for example:

sudo mkdir /mnt/partition_penguin
sudo mount /dev/sda8 /mnt/partition_penguin

Bind in the system directories to give access to the hardware:

sudo mount --bind /dev /mnt/partition_penguin/dev
sudo mount --bind /proc /mnt/partition_penguin/proc
sudo mount --bind /sys /mnt/partition_penguin/sys

And then enter the chroot environment and run your commands:

sudo chroot /mnt/partition_penguin /bin/bash
sudo aptitude update