There are two ways to do this, depending on whether the user's role is a Web Administrator or Content Publisher:
For Web Administrators
If the user is a Web Administrator, then they will not automatically receive ANY Ready to Publish notifications. To override this, go to the Configuration tab and click on Advanced Settings>Application Settings. Select the Settings Related to Email category and find the option called PublishNotificationEmails. Put the e-mail address of users that should receive in this field and they will immediately begin receiving all Ready to Publish notifications generated by the system. Multiple emails, separated by a comma, can be added.
For Content Publishers
If the user is a Content Publisher, then it is only necessary to make sure that they have publish rights to all pages in the system. On the Users tab, find the user, click the gear icon next to their name, and select Permissions. On the Pages tab find the Can Manage Any Page permission and make sure that it is set to Yes. This will grant the user permission to publish any page in the system, and therefore the user will receive all Ready to Publish notifications. If the user only has permission to manage specific pages, they will receive Ready to Publish notifications when a Staff member or Content Editor sets the page as Ready to Publish.