Active Blog while developing WP Site

I have a client who has an existing website, but it is old and non mobile compliant. I am developing a new WP site for them. They want to start blogging right away. I’m pretty sure I know how to link the blog to the old site (while the new site is being developed), but I’m not sure how to “isolate” the new blog page so that the new site menus don’t allow access to the new site until it’s ready to launch. I’ve tried some plugins that are supposed to hide the menu on a certain page, but I couldn’t get them to work right. I want to have the blog page “live”, but I want to still be able to work on the rest of the site and view my changes as the site develops.