While the title says Raspbian on Ubuntu, the steps will be nearly the same for any ARM linux distribution on any AMD64 Linux desktop. For me, and I suspect most people, what combination is likely to equal Raspbian on Ubuntu.
Download and extract your Raspbian image. In this case I will be using Raspbian Lite and this is the filename of the image:
sudo apt install qemu binfmt-support qemu-user-static systemd-container
After the setup, run the following steps every time you want to run a Raspbian image on Ubuntu.
sudo kpartx -v -a ./2019-04-08-raspbian-stretch.img
That will print something like:
add map loop11p1 (253:0): 0 89698 linear 7:11 8192
add map loop11p2 (253:1): 0 8392704 linear 7:11 98304
The first line is the raspbian boot partition, the second will be the rootfs partition.
udisksctl mount -b /dev/mapper/loop11p2
After the prior command, a rootfs mount should appear in /media/<your_username>/. You can copy files into either and they will be placed into the image file you mounted.
sudo systemd-nspawn -D /media/<your_username>/rootfs/
Ctrl-D to exit.
udisksctl unmount -b /dev/mapper/loop11p2
sudo kpartx -d ./2019-04-08-raspbian-stretch.img
Note, a more advanced use might be to copy some sort of initialization script into the image after mounting the image and before running systemd-nspawn. Then, invoke that command something like:
sudo systemd-nspawn -D /media/<your_username>/rootfs/ /bin/bash /home/pi/your_script.sh