You can configure WordPress to have plugins in a different location. Post Types Order doesn't respect this setting. The fix is trivial:
At the top of the plugin file, replace:
define('CPTPATH', ABSPATH.'wp-content/plugins/post-types-order');
define('CPTURL', get_option('siteurl').'/wp-content/plugins/post-types-order');
with:
define('CPTPATH', WP_PLUGIN_DIR.'/post-types-order');
define('CPTURL', WP_PLUGIN_URL.'/post-types-order');