By default, standard WordPress installs give the option to perform automatic updates using FTP or FTPS (FTP over SSL). There is another method to have Apache perform the update without passwords. It can be activated by adding the following to the wp-config.php file:
This does require that you have the permissions set up correctly, and you should be aware that there are serious security implications doing this.
Details on the settings for wp-config.php can be found in the documentation Editing wp-config.php.