Microsoft Office 2016 update button missing

You are using the click-to-run version of Office 2016. Thus, the Office updates are not controlled by Windows Update.

When automatic updates are enabled for Office clcik-to-run version, in most cases updates are applied automatically in the background without any user input. However, updates can't be applied if an Office program is open. If an Office program is open, other attempts are made to apply the updates at a later time. If, after several days, updates haven't been applied, only then will users see a notification that an update to Office is available.

To disable the update notification, we can try the following registry key settings:

HKEY_LOCAL_MACHINE\software\microsoft\office\16.0\common\officeupdate
Value Name: hideupdatenotifications
Value Type: REG_DWORD
Value Data: 1 

You can also create the hideupdatenotifications value with group policy setting to disable it:

HKEY_LOCAL_MACHINE\software\policies\microsoft\office\16.0\common\officeupdate

I had this issue as well where I had a Visio 2013 (.vsdx) and it would not allow editing of embedded Excel files. So I had to convert one of two ways: 1. On a machine with Office 2013 save the file as an older version on Visio (.vsd) and then open it on the Office 365 machine and it works fine. Also, you will be able to save the file back to a .vsdx file on the Office 365 machine and open and edit it normally.

It appears that once you save it as a .vsd file on 2013 environment, then open the file, edit it on 365 environments and then save back to the .vsdx file. After this, the file can be edited normally on the Office 365 machine.

It appears Office adds all the versions to the file so that it is now editable in any office environment (2013, 2016, 365). But you have to start with saving the file as .vsd on an Office 2013 machine.

The issue is that Office 365 and Office 2016 apps do not work well on the same machine.