First, keep in mind that with e-mail there are no guarantees. The SLA is a best effort one and that is it. In fact, it is a wonder and a tribute to the designers, that e-mail works as well as it does.
First question to be answered is “Was Outlook or whatever mail program you are using, up and running to send the message on January 1?” If not, you have your answer right there. The message was sent when the program was loaded on January 2. If it was running on January 1, then there are a few methods you can use.
The definitive one is to have the recipient send you the complete headers from the message. From the headers, you can gain all the information you need to arrive at an explanation.
One tool you can use is message tracking, which will show what time the message passed through your server. If it left your server on January 1, you are off the hook.
The other method would be to peruse your SMTP logs. This will show the conversation with the first receiving server along with the time each back and forth between the servers occurred. This will show whether there was a problem in communications between the two servers, whether the mail server on the other end was down and the message sat in your server queue for sometime before being sent, and other nifty stuff like that. This method can leave you a bit bleary eyed and perhaps dizzy, since there could be much talk between many servers about various messages being sent and received so that you would not find all the information about the message in question in one neat block. I do not do this work very often so I cannot recommend any parsing tools that may help you, but I am sure that someone else can chime in to recommend their favorite, or just tough it out to find what you need (find can be a very good friend here).
If you have asked, and the recipient has told you that there was no problem with their e-mail over the weekend, I’d take that with a grain of salt. The admin could have already resolved the issue before most people may have been aware that there was a problem. If this was the case, your server would have tried to resend your message on a periodic basis. When the server came back online, there would have been a backlog of messages to process and his server would have started. Your message would have been sent at the next opportunity, and may have hung around in his server waiting to be processed and placed in the recipient’s mailbox.