If you are being asked for the FTP credentials, check the Upgrade constants that you can include in your wp-config.php to bypass that request.
To upgrade a plugin manually, it's a matter of uploading its folder to your server in the following path: /wp-content/plugins/.
And, in the manual process, uploading and/or deleting a folder can take some time. And, if the plugin is active, this could lead to a temporary break or malfunction of the site. So to make it transparent as possible, use the following technique.
- Create an auxiliary folder
/wp-content/to-delete/
- Upload your updated plugin directly in
/wp-content/
- When finished, move the old plugin (
/wp-content/plugins/the-plugin/) to the auxiliary folder
- Immediately, move the updated plugin folder (
/wp-content/the-plugin/) to the plugins folder
- Delete the old plugin folder (
/wp-content/to-delete/the-plugin)
All this assumes that you understand WordPress file/folder structure.