I haven't tested this on Mint specifically, but generally you can use /etc/sudoers (using visudo) to specify which users can run which commands.
An entry along the lines of:
- Code: Select all
alice ALL=NOPASSWD: /usr/bin/mintupdate
...should let alice run mintupdate with escalated privileges, without being prompted for a password.
As usual, the Arch wiki is good reading: https://wiki.archlinux.org/index.php/Sudo#Configuration