If you setup a schedule task, it will run and update it, yes.
The schedule task procedure is described at the bottom of the link you posted above:
Configure cron so that the certificate will automatically renew every month.
crontab -e
In the following line, use your own domain name in place of {domain-name}
@monthly /home/ubuntu/letsencrypt/letsencrypt-auto certonly --reinstall --nginx -d {domain-name} && sudo service nginx reload
Make sure you replace the paths with the ones you have used in your installation.
And since you don’t have nginx or apache you can skip this later part of the above command:
&& sudo service nginx reload