Hi,
I've never tried but I think it can be done.
You would need to create a custom field of the table "product" and then a mass action with an action "update the values".
In the action, you would select the column of your custom field, and the mode "operation" and then enter in the input field:
category.category_name
That way, when you process the mass action, it would fill the custom field with the name of the category, and you could then use the display settings of the custom field to display the information in the checkout, invoice, emails, etc.