Kernel Panic - not syncing: VFS: Unable to mount root fs on unknown-block(0,0)

You are missing the initramfs for that kernel. Choose another kernel from the GRUB menu under Advanced options for Ubuntu and run sudo update-initramfs -u -k version to generate the initrd for version (replace version with the kernel version string such as 4.15.0-36-generic) then sudo update-grub.

Start with a livecd, open a a terminal

sudo fdisk -l
sudo mount /dev/sdax /mnt
sudo mount --bind /dev /mnt/dev
sudo mount --bind /dev/pts /mnt/dev/pts
sudo mount --bind /proc /mnt/proc
sudo mount --bind /sys /mnt/sys
sudo chroot /mnt 

and now you can make update-initramfs and update-grub without errors.

update-initramfs -u -k 2.6.38-8-generic (or your version)

If you don't know your version. Use:

dpkg --list | grep linux-image

And just update Grub.

update-grub2

Reboot your system.

In case this happened after an aborted kernel update (e.g. system crash while aptitude safe-upgrade),

1. boot with an older kernel and
2. run dpkg --configure -a.

This will complete the upgrade, including configuring the boot settings as psusi explains.