How can a document library in On-Premises SharePoint 2013 but kept in sync with a document library in SharePoint Online automatically?
The only way to push data into SharePoint online is via either CSOM or REST APIs. Either way, it's a client side tech. Since your source is on prem, you can use event receivers (DocumentUpdated) as the trigger which would push the update via the client side tools to 365. Start with event receivers here: https://msdn.microsoft.com/en-us/library/ee231563.aspx Start with CSOM here: https://msdn.microsoft.com/en-us/library/ff798388.aspx If you prefer REST, start here: https://msdn.microsoft.com/en-us/library/office/jj164022.aspx