You should be able to use Qemu User Emulation to accomplish this via
chroot. Its a similar sort of setup to debootsrapping a foreign arch but you'll already have the OS installed.
This assumes your host is Debian based - i'd assume the Qemu stuff is available on other distros though.
Important steps from the Debian Wiki
sudo apt-get install qemu binfmt-support qemu-user-static
Mount your Pi SD card somewhere on your linux host
sudo mount /dev/mmcblks0p1 /tmp/pizero (or whatever it is called on your system)
Copy the Qemu binary onto the pi SD card (can be deleted when you've finished) - might not need to do this anymore though.
cp /usr/bin/qemu-arm-static /tmp/pizero/usr/bin
Chroot into the Pi and you can then treat it as if it was running natively.
sudo chroot /tmp/pizero
and you should end up as root on the Pi and able to run apt and stuff against the Pi FS.