Normally after you create an email account using cpanel, you should be able to use the following info:
email acct: email@example.com
server incoming: mail.linaria.gr sometimes requires port 25
server outgoing: mail.linaria.gr sometimes requires port 26
In the global configuration under mail, you do not have to set the SMTP info normally if you are using the php mail function setting.
If you wish to then use the info like that above and set the Mailer to SMTP server.
The SMTP user is firstname.lastname@example.org
in your case.
The SMTP host is either localhost or the domain name, linaria.gr in your case.
As far as what you say about the email account going through google apps, I don't know about that. I would do that anyway because it is just as simple to forward emails to another account through cpanel. I think that if your client is using google mail then he/she might be getting it through the google mail options to receive pop mail from another account.
As a simple test just send an email to yourself from the website.