We are an educational institution. Recently we changed our traditional academic licensing model of Microsoft Desktop Education Suite for the subscription of Office 365 E3 which includes Office 2013 ProPlus.

We need to update 400 machines from Office 2010 to Office 2013 without having to run the Office 365 desktop setup. We want to install the ISO in each machine and activate the Office 2013 with one product key.

What would be the proper way to perform this implementation?

There is no single correct answer to this question. I would suggest looking at the Office 2013 deployment options available with System Center 2012 R2 Configuration Manager or App-V https://gallery.technet.microsoft.com/office/How-to-Deploying-Office-0f954e7f https://technet.microsoft.com/en-us/library/dn817830.aspx