Source: https://todhost.com/blog/wordpress-manual-update-tutorial.html

Updating WordPress manually is an option especially when you do not have admin access to automatic and you want to quickly fix things up. Manually update is an option you have. There are several reasons you may not be able to update WordPress to the latest version automatically or you may want to try out the latest Beta version which doesn't offer automatic updates at this time.

WARNING: The WordPress update process will affect all files and folders included in the main WordPress installation. This includes all the core files used to run WordPress, the Akismet plugin and the WordPress default theme. If you have made any modifications to those files, your changes will be lost.

Befoe you get started
• Backup your WordPress website. Just incase things do not turn the way you desired, you can still be able to roll back with your backup.
• Check New WordPress Requirements before updating
• Check server requirements to be sure you are running compatible server specifications including PHP and mySQL versions.
The requirements have changed as of WordPress 3.2.

Recommended for better security - suPHP: This is not a requirement however we do recommend you use a host which offers suPHP for increased security. Todhost has suPHP installed on server.

Check Plugins
If you're updating WordPress before your plugin authors have tested their plugins with the latest version of WordPress then you could run into trouble. Installing the Beta version of WordPress would mean most, if not all plugin authors haven't had a chance to test against the latest WordPress update yet.

Manual WordPress Update
First step:
Download the latest version of WordPress here http://wordpress.org/download/
• Unzip the files: Delete your existing old wp-includes and wp-admin directories on your server (Using FTP or File Manager).
• Upload the new wp-includes and wp-admin folders (directories) to your server.(using FTP or Shell Access SSH)
• Open the wp-content folder and upload the individual files from the new folder to your existing wp-content folder (Using FTP, File Manager or Shell SSH), overwriting existing files. Do NOT delete your existing wp-content folder. Do NOT delete any files or folders in your existing wp-content directory (except for the one being overwritten by new files).
• Upload all remaining new individual files from the root directory (folder) of the new version to your existing wordpress root directory.

Caution: If you have customized your themes without renaming them, make sure not to overwrite those files, otherwise you will lose your changes.

Second Step:
Update WordPress by logging in to your dashboard and follow the update instructions. You will probably need to upgrade your database simply by following the process and clicking upgrade.

Third Step:
You may need to Clear your cookies, Delete or refresh your cache if you’re using a caching plugin like WP Super Cache or W3 Total Cache

Re activate your plugins if you deactivated them before upgrading WordPress (recommended)

Your update has been successful and you can now play around with the latest version of WordPress.
Remember: To avoid losing any custom changes you make to the WordPress core files after you update, you may consider creating your own child theme which will save all your custom coding in a separate directory and therefore not be overwritten when you update to the latest version in the future.

Source: https://todhost.com/blog/wordpress-manual-update-tutorial.html