Emails coming in late (delayed emails)

Two questions.
1. If you don't trust your IT Manager, why are you retaining his services?
2. So you don't trust your IT manager, who is presumably a trained professional, with direct access to the system for analysis, and our instead going to trust some random strangers off the internet, most, if not all of whom, have no professional training on this whatsoever, and in any case have no access to the system for analysis. Why is he not looking for another job?

Well, I am having some trouble following what the IT manager is saying but I'll do my best. Please note it is possible that the IT Manager is trying to explain something in "laymans terms" and thus, did not give you the complete, accurate response. With that said, the response doesn't make sense.
Outlook synchronization is used to synchronize an offline\local copy of your mail file with the "live" copy of said mail file on the exchange server. Synchronization errors do happen, and are actually pretty common (though you normally don't see them as they usually don't hurt anything). However that has nothing at all to do with when an email is delivered. Assuming a major sync error occurs that completely prevents the ost (local mail file) from being updated, the email is still delivered to the mail box. The user, in this case the COO, would not see them in outlook, but they would be there in his mail file. And, when he goes to OWA, that pulls directly from the exchange server and he would see them there. If he isn't seeing them there either, then the mail has not already been delivered and synchronization has nothing to do with it.
My guess based on what was written is that a major problem occured (i.e., an outage, affecting at least the exchange server the COOs mailbox is on) and this prevented the mail from being delivered; it may have also caused the separate issue of a sycn error. So I wouldn't say he is lying per say, just either a. downplaying what is essentially a mail outage (big issue) and calling it a sync error (small issue), or b. trying to explain something in a way that he thinks you will understand.
As I said at the begining though, this is all guessing based on what you wrote which may or may not be accurate since I am getting it third hand.

Well, I can't really say if those steps are appropriate because I don't have access to the system, the logs, etc. So I don't know what is the root cause of the issue. Unfortunately, he is the only person who probably has enough information to make that determination. With that said, while I have no idea what equipment you guys are using, assuming it is even remotely modern, telling the end users to reboot periodically throughout the day is not a reasonable solution. Neither is forcing users to perform manual synchs regularly. I will say that if one of my vendors\service providers told me that the solution was for everyone to periodically restart their computer, the contract would be terminated immediately.