Tag Archive for Custom-Post-Type Posts yielding 404 when permalinks set to postname

Try give a name for your rewrite, like so:

'rewrite' => array(
    'slug'       => 'news-updates-archive',
    'with_front' => true,
    'feeds'      => false,
    'pages'      => true,
)

So you will avoid any problem.

Also, the last functions you’re using are unnecessary. You can setup it on register_post_type:

'taxonomies' => array('post_tag', 'category'),

PS: Try empty your trash also and double check if your custom post type name is not duplicated. Sometimes that’s the error.