I don’t think there is an API in Revit that would allow you to access that history of synchs. I can think of creating your own plug-in for Revit that generates such history but that would require coding up an add-in for Revit and hooking into
DocumentSynchronized event. That’s pretty easy to do. Then you would just need a place to store it. A database would do. All in all it would have been possible to create such a startup view, but I don’t think that Dynamo is the right tool for it. With a little bit of coding you could automate that COMPLETELY so users wouldn’t have to run anything and that view would just update.
One thing to consider: Are you sure that it’s a good idea to invest this much time into creating a startup view? What’s the return on that investment besides a view that presents information to the end user, that frankly, I doubt any end user would pay any attention to. Just my two cents.
Ps. I couldn’t find any documentation for getting the history in Revit API, but if someone else has more luck please share here. I will be happy to stand corrected.