You could setup a crontab to use tmutil through the command line to do a backup and then shutdown. since shutdown requires sudo privs, you have to set your crontab up as sudo.
sudo crontab -e
Enter your password and then that brings you to edit the contab file
shutdown -h now
Shutdown does exactly that but with the -h flag, it halts the system (shutdown).
For the shutdown to work, this would have to be setup as an crontab under sudo
Starts the timemachine backup.
Putting it all together:
00 20 ** ** ** tmutil startback --block; shutdown -h now
Would do thebackup at 10:00pm every day and then shutdown