If I got you correctly, WP Theme Test is such a plugin that can assist you.
In my case I take the whole live site in localhost, by copying the whole WordPress installation and its database, and then test any new theme on it (or any modification to existing theme). If the local test passed, then I go for the live site test. And for a live site theme testing, plugin like WP Theme Test is very helpful. (There’s a plugin called Theme Test Drive, but the plugin retired recently)
Without taking the whole site into maintenance the plugin will activate the new theme only for logged in administrator user[s]. So you can check all the things of that theme in your live site without leaving any trace to any visitor or any other logged in users (other than admins).
So if everything shows alright, you can disable the Theme Test mode and disable the plugin too; and you are green to activate your new theme to the live site and configure it as it needs.
For the transitional period when you are going to change the theme, plugin like Easy Maintenance Mode is very helpful. You can take the whole site in maintenance mode from visitors with a nice message and social network’s links and with Google Analytics code, so the site will be safe for any visitor (because they will notified with the message and will be in touch with the social links), and for Google crawler (because you will still get your analytics). After completely setting up your new theme, you can disable the maintenance mode, and make the site live again, with a big smile. 🙂
But always take backup before a big change, because if anything go wrong, you can then have a restore point to back to. 🙂