Any links you’ve manually created will not be automatically updated. This includes links in the navigation if you manually created them as direct-entry URLs.
There’s really no difference or improvement in moving while upgrading or just moving.
An .htaccess rewrite would do a fine job of making old links work without you having to fix them, if that’s the route you want to go. This is getting a little beyond the scope of Omeka support and into Apache configuration but it should be as simple as
Redirect "/omeka" "/new_name"
in an .htaccess file in your web root folder (or Apache config or virtual host config or wherever).