Hooks are only available when you see apply_filters, apply_filters_ref_array, do_action, and do_action_ref_array. They are all basically handled the same way but have some small differences like filters always expecting a returned value.
e.g.:
// seeing this in some WordPress code
$the_content = apply_filters( 'the_content', $the_content );
// means you can then use something like this to modify it
add_filter( 'the_content, function ( $content ) { return $content; } );
The method you refer to has none of those function calls directly in it, so you will need to find a place, either up or down the call chain, where a filter is applied that relates to what you want to modify.
You may be better to ask how to achieve your ultimate goal rather than asking about how to make this work as there aren’t any filters or actions in WP_Embed::run_shortcode or the do_shortcode it calls or the do_shortcode_tags and do_shortcodes_in_html_tags.
For example, there is a filter for the embed handler html output that might be enough for your needs:
...
return apply_filters( 'embed_handler_html', $return, $url, $attr );
...
which would mean you could do:
add_filter( 'embed_handler_html', 'wpse_202291_embed_handler_html', 1, 3 );
function wpse_202291_embed_handler_html( $return, $url, $attr ) {
// process $return
return $return;
}