How to update Drupal 8 Core manually

Using CPanel

  1. Log into your site, put it into maintenance mode ~/admin/config/development/maintenance and back up all files.
  2. Log into cPanel and start the cPanel file manager.
  3. Go to the root directory, usually /public_html.
  4. Delete all Folder & files except Modules, Profile, Sites, Themes
  5. Upload latest drupal version - example: Drupal-8.4.8.tar.gz and extract it.
  6. Copy all files and folders EXCEPT: Modules, Profile, Sites, Themes folder to the root directory.
  7. Log into your site and run database updates ~/update.php
  8. Check the Status report ~/admin/reports/status.
  9. If all OK take the site out of maintenance mode

Source: https://www.drupal.org/forum/support/post-installation/2016-03-08/how-to...