I believe you will have to make sure the task is run as the root user or else it will silently fail when you don't supply the password.
Save the following attachment in your home folder. Extract it from the zipped compression (right-click and "Extract Here") then open a terminal and run these two commands to a) move the upgrade script to the executable directory and b) make the script belong to root:
Code:
sudo mv calibre-upgrade.sh /usr/bin/calibre-upgrade.sh
sudo chown root:root /usr/bin/calibre-upgrade.sh
Then run this:
Code:
sudo EDITOR="gedit" crontab -e
This will open the root crontab in the
graphical text editor.

I figured this is not the time for you to start learning nano or (heaven forbid) vim.
EDIT: Because vim is a complicated sysadmin tool, not because its a bad editor.
paste this line on the bottom,
on a new line:
Code:
0 6 * * 5 /usr/bin/calibre-upgrade.sh > /dev/null 2>&1
Then press enter, as crontab must end with a newline or it goes kablooey.
Save the file, and it will update your crontab and install the auto-update code.
Install "anacron" in the software center, and missed jobs should be carried out shortly after you reboot.