Can I bind a post_type=’post’ category to a custom post_type category?

If you need to register the built in category taxonomy for a custom post type you can do this by using the priority argument when adding an init action hook.

add_action( 'init', 'add_category_to_post_types', 11 );
function add_category_to_post_types() {
   register_taxonomy_for_object_type( 'category', 'my_post_type' );
}

The third argument in the add_action() call sets a priority so most things in wordpress will occur at the default priority of 10. By setting this to 11 your custom post type should be registered and the built in taxonomies will definitely be registered so you can add to your post type.

If that doesn’t work you need to check the priority of the action hook in which your post type is registered and set the number 11 above to something higher than that.