Sorry, way too late to get the "free upgrade". Here is a link to the dead "offer" page:
Offer Expired page dead
You can try reinstalling Visio 2013.
Office 365/2016 is totally unreasonable about installing alongside older versions, forcing us to uninstall them, but 2013 does not (yet) have that unreasonable bias added. I THINK it will install. (This page says that won't work
https://technet.microsoft.com/en-us/library/mt422981.aspx
... but give it a try anyway)
Supported scenarios for installing different versions of Office, Visio, and Project on the same computer–
Different Years and MSI vs CTR.
https://technet.microsoft.com/library/mt712177%28v=office.16%29.aspx?f=255&MSPPError=-2147217396
This is an EXCELLENT resource.
Starting on October 11, 2016, Office 2013 software that uses Click-to-Run can be installed on the same computer with Office 2016 software that uses Click-to-Run.