Update module to dev without Drush
Drush is a very useful tool and it will take some tasks from your hands, including running update.php after updating a module.

However if you cannot (or do not want to) use Drush, there are plenty of other options to update modules. You can use FTP to delete the module folder and then upload a new version of the folder. You can use SSH to obtain the module zip file by using wget and then delete the module folder and extract the zip file. Basically any possible way to get the contents of the zip file to your destination is a valid updating method.

Keep in mind that you cannot always safely unzip one module directly over another one. Always delete or empty the previous module version first. The reason for this is that several (class) files might have been removed from the module, but would not get removed from the file system this way. Drupal's registry might still load these files but then find another class with the same name in another file, resulting in fatal errors.

And no matter how you update the modules files, always stick to the regular updating instructions. This includes always running update.php.