update post category in a new table on frequent change of category

Question

You could use an Action Hook ‘save_post‘ to implement the requirement and this way categories will stay synced with your custom table and default tables.

Example :

 function post_updated_sync_category( $post_id ) {

    //Here you get postid of the post being updated
    //You can put your code to update in custom table here
}
add_action( 'save_post', 'post_updated_sync_category' );

You can use this hook with custom post type as well, here is how you can implement this, see WordPress Codex