If you want to easily update Drupal core, then this is right for you. This approach will first compare your deployed Drupal with the original Drupal source code to do a pre-update sanity check. Then the old Drupal files are deleted one by one, the empty directories as well, and then the new Drupal is deployed.
If you follow this approach, updating your Drupal core version should take you less than 5 minutes.
If you have no shell access to your hosting provider, you can still apply the approach described in this article if you have a mirror copy (a perfect replica) of the deployed Drupal server on a host you have shell access to. Apply the changes on the mirrored files on your local host (steps 1 to 5, skip step 6), then synchronize the changes with the webhost, and finally run update.php (step 7).