Yes - the NS Cloner can do that, or really close to it. After installing the plugin, you can go to your multisite network admin page (http://yoursite.com/wp-admin/network), find the Cloner page in the network admin menu, and then just follow the instructions there to create a new sub-site on your network which is an exact copy of the main site (after which you could modify/replace the main site's content and settings without affecting the clone at staging.domain.com).
Note that this wouldn't technically be exactly the same as creating a new main site from scratch, due to the way that WP multisite works there always has to be a main site, but I think would achieve the result you're after. Also probably goes without saying but keep in mind that WP sites on a multisite network share the same theme and plugin files, so if the staging site and main site have the same theme and plugins activated, editing those files will affect both sites.
The cloned staging site will have the same theme and plugin settings, pages, posts, custom post types, media attachments, etc. The only thing that the free version does not do that you mentioned is to copy users. For that you would need the Content and Users add-on. But if you don't have a real large number of users you could still just go to Network Admin > Sites > ["Edit" link below the new cloned staging site] > Users and then transfer the users over manually by typing in their user names in the "Add Existing User" section.
Hope that helps!