We are using SharePoint 2013 and planning to customize the document upload window as shown in the picture. 
This is a customized application page. Could anyone advice me whether this kind of customization is supported in office 365 if we move the solution to cloud based office 365 Enterprise version?
You cannot customize upload.aspx in Sharepoint Online directly.
There are couple of workarounds for doing so:
No, you won't be able to move that to Office 365. You'll need to recreate the design entirely in JavaScript. You can deploy a custom action in a sandbox solution that will allow you to hook up a JS reference, and then render your markup there.
Now, I assume you actually want to do something with those checkboxes. That's a different story, and the more difficult one if you ask me.
