Although you will need to change some permissions to make the web based update work.

The next TurnKey version to be released (v14.1 - coming really soon hopefully) will have WordPress v4.4.x (although TBH I'm not 100% on exactly what version).

To do that try this:

chown -R www-data:www-data /var/www/wordpress

Hopefully that should be all you need to do. If it still complains about wanting FTP credentials (or something similar) then please double check that you have this in your config (IIRC should be /var/www/wordpress/wp-config.php):

define('FS_METHOD','direct');