Yes dashboard is available in publisher instance. But few of the options are disabled in publisher.
You can access the dashboard with publisher port. For example if your publisher instance is running on 4503 port you can access with same url by changing the port number. http://localhost:4503/aem/start.html
When you start AEM in publish mode, there are features that are disabled (such as the editor), however, you should be able to still access the touch UI interface from http://localhost:4503/aem/start.html, make sure you are logged in before visiting the touch UI interface.