Upgrading Drupal 7 version from DEV to PROD
Even thought you said that you are not comfortable with Drush. I would strongly suggest for you to used Drush to update your modules and core. Its just easier and faster. If you decide to give  a try you can update modules and core by running.

drush up -y


If you want to update just the core use drush up drupal -y

You can upload a copy of the latest Drupal version on your production website and then replace sites folder with your dev sites folders. And this would take care of Drupal Core updates.