From a live disc, it won't do what you want it to do. However, quoting from the
man page:

"update-grub is a stub for running

grub-mkconfig -o /boot/grub/grub.cfg

to generate a grub2 config file."

So, if all you need to do is regenerate grub.cfg (i.e., your MBR is OK) you can
boot from the Live CD, mount your normal boot partition (I'll assume as /mnt)
and run these commands from terminal:

cd /mnt
cd boot/grub
sudo cp grub.cfg grub.cfg_BAK
sudo grub-mkconfig -o ./grub.cfg

(Caveat: I haven't tested this.)

If you *do* have to repair the MBR, use the instructions in this link from
someone else's post:

https://help.ubuntu.com/community/RecoveringUbuntuAfterInstallingWindows