Cloud tools are nice, but it's a little bit too many features and expensive for non-profit use, that is why I just try to do this simple thing on exporting them on local share so others can access it.

How do I connect this Print button to a controller method? Let's say when we press Print that our method adds extra functionality to save it in folder and then call the super class method?

The point is that when you press 'print', Odoo already creates an attachment (ir.attachment instance) or use already existing one. So, you need either to find the attachment on your server as using the first link above; or to find it in Odoo and proceed your actions. If you want to re-write printing methods look at 'invoice_print' of account.invoice or 'report_action' and related method of reports