Redirect any emails that are sent by the staging site to the administrator of the site.
On a normal WordPress installation, go to the Dashboard, then to Settings > General. Look for the "E-mail Address" field.
On a Multisite WordPress installation, go to the Network Dashboard, then to Settings > Network Settings. Look for the "Network Admin Email" field.
It is recommended to activate this plugin on your Production site. It won't have any affect on your Production site email, and then when you go to recreate your Staging site, it will already be activated and working.