The latest Office 2013 build (as of May 2019) is “15.0.5127.1000”. This is required to connect Outlook 2013 to Office 365.
If you have an earlier build, you may find that Outlook 2013 does not accept the Office 365 Password.
Finally, if you find that the package will not auto-update to the latest build, you may need to perform a “Online Repair”. You can do this from within: Windows Control Panel => Programs =>Programs and Features => Office 2013 => Online Repair.
Microsoft Office 2013 and Exchange Online
Practical SBS 