Do you have root access on your 3rd party hosting service ? and does this have some sort of control panel which manages the apache / nginx configuration ?
if it’s a standard server, running apache, you have root access ( i.e. permission to modify the apache/nginx configuration) then yes you can run letsencrypt-auto. If it has say a control panel, such as cpanel, plesk, ispconfig then you can’t just run that command (hence why I said it’s difficult to give a definitive answer without that info).
For checking when your certificate expires, if you use linux, I use a script ( checkssl ) but you could just manually check the certificate in your browser ( click on the lock, and go to certificate information )