You do not have to do any of that.
The difference between
www.meccanicaprealpina.it/helpdesk
and
helpdesk.meccanicaprealpina.it
are controlled by host records, dns, and your webserver site config.
And then once you have updated those things, you would change the Helpdesk URL at Admin panel ->Settings -> System
*** TLDR answer ***
If you want the site to be reachable out outside of your network then you should probably setup a entry for it in your host record for your domain (meccanicaprealpina.it). To do this you would add an A Record for helpdesk and point it to your webserver.
If you run your own DNS servers you would do the same thing internally. If not then you might need to wait for DNS to propagate from the host record change. If you can do a nslookup on helpdesk.meccanicaprealpina.it and get the correct answer then you are good to go.
I'm not sure what webserver that you are running (since you didn't tell us) but in IIS you would want to create a binding for the site for helpdesk.meccanicaprealpina.it. If you do not know how to do this, then I would recommend that you search for a guide.
If you are running Apache then you could edit the config for the site and add a ServerAlias to the config file and restart the site.
it might look something like this
<VirtualHost *:80>
ServerName www.meccanicaprealpina.it/helpdesk
ServerAlias helpdesk.meccanicaprealpina.it
DocumentRoot "/www/domain"
</VirtualHost>
note: this is an example not a working config file.
