Is it possible to hook AJAX to UPDATE-button?

If all you’re wanting to do is save extra post data you should not need jquery or custom ajax. WordPress has a built in action you can use to achieve this same thing.

add_action( 'save_post', 'save_more_post_meta' );
function save_more_post_meta( $post_id ) {
    //save stuff here
}

If you need help with saving specifics i’ll need a little more detail