Since it's Outlook, the first place to check is the Outbox. If the mails are leaving the Outbox, then that means the computer is set up correctly for sending, and that's where your friend's involvement ends. If an ISP is genuinely unreliable - that is, if the problem is not on the user's side - there's little that an individual user can do about it, other than seek alternative.
Switching to a web-based email provider would be a start. I think Eircom does have a web interface to their mail system, so that might be the first thing to try, but if that doesn't improve matters: Gmail.