If your user is called user and your host is called you could add these lines to /etc/sudoers:

user = (root) NOPASSWD: /sbin/shutdown
user = (root) NOPASSWD: /sbin/reboot

This will allow the user user to run the desired commands without entering a password. All other sudoed commands will still require a password.

The commands specified in the sudoers file must be fully qualified (i.e. using the absolute path to the command to run)

If the command ends with a trailing / character and points to a directory, the user will be able to run any command in that directory (but not in any sub-directories therein). In the following example, the user user can run any command in the directory /home/someuser/bin/:

user = (root) NOPASSWD: /home/someuser/bin/

As an alternative to editing the /etc/sudoers file, you could add the two lines to a new file in /etc/sudoers.d e.g. /etc/sudoers.d/shutdown. This is an elegant way of separating different changes to the sudo rights and also leaves the original sudoers file untouched for easier upgrades.

visudo can be used to edit those files too, this prevent error that could lock you out of the system

sudo visudo -f /etc/sudoers.d/shutdown