I also use Outlook 2013 and EAS on a Surface Pro 3. There are certain scenarios where I experience similar issues.
Unfortunately, all of them are related to quirks in Outlook 2013.
For example, when a computer goes into sleep or hibernation and then wakes up, Outlook does not re-establish the connection with the server. If you hit the Send/Receive Email, everything will work again.
We have have also seen Firewalls setup to drop connections after X amount of time. Where phones are really good at re-establishing a connection, Outlook is not. Again, if you hit Send/Receive Email it starts working again.
When Outlook does its initial pull of all emails from the server using EAS, Outlook does not do a good job handling work that a user is doing at the same time. For example, if I pull down my entire mailbox which was the DEFAULT setting for Outlook 2013 and EAS (which is INSANE) because my mailbox is MANY GIGS... If I delete a new email that comes in or move messages in a folder that had already synced but other folders are still syncing... Outlook may or may not actually send these commands to the server when it has free time or finishes with the initial sync. It is as if Outlook forgets about the queue of work that the user has done. Unless Outlook communicates the requests to delete or move these emails, we can't do anything about it. Anyway, we really encourage customers on initial syncs to leave their computer alone until its done.
My guess is, your user can hit Send/Receive Email and it will work again? If so, then you know its a connection being dropped for one reason or another and Outlook isn't good about fixing it.
One solution could be is that your customer could simply change his power savings mode while plugged in to not Sleep the computer. That could help a TON! He should look at his connection and make sure its not going in and out.
There are other things to look at, these are just the more common and things that I have specifically ran into or helped people with. You can always open a support ticket and we can take a look at your server and logs. We can give you a better idea if Outlook isn't connecting or no longer has a connection to the server etc.
Hope this helps,