How to moderate (manually approve) comments of a specific (registered) User

If you are comfortable with coding, you could try a custom filter based on the WP_Comment object. Perhaps something like the following:

function wpse_wp_insert_comment($id, $comment) {    
        // Add your user_id, or use email address instead.
        if ( empty($comment->comment_author ) || $comment->user_id!== 1 )
            wp_set_comment_status( $comment, 'hold' );
}
add_action('wp_insert_comment', 'wpse_wp_insert_comment', 10, 2);

-Or …

// Same method, but using userdata to compare email address instead
function wpse_wp_insert_comment($id, $comment) {    
    $author = empty($comment->comment_author) ? NULL : get_userdata($comment->user_id);
    // Be sure to update this line with YOUR actual email address.
    if ( empty($comment->comment_author ) || $author->user_email!=="[email protected]" )
        wp_set_comment_status( $comment, 'hold' );
}
add_action('wp_insert_comment', 'wpse_wp_insert_comment', 10, 2);

Extrapolated from a Tom Mcfarlin article “Programmatically Mark a Comment as Unapproved“. See the “Gotcha” warning he throws out regarding users’ anticipated outcome vs actual manipulated results caused by this type of function.