The build server can run the unit tests and even UI tests for your app. For UI tests, think of tools such as Espresso (Android) or Xcode UI testing (iOS). If you are looking for tools that support both platforms you can check out Appium, for example, (available for Android and iOS).
When it comes to testing, there are different approaches to consider. One of these approaches is Test-driven Development (TDD). If the functionality and all the requirements are known, we can define our tests before developing our app. Of course, all tests will fail initially, but that is actually a good thing because it will set an outline of what needs to be done and create focus to get things right. During the implementation of the features, your tests are going to succeed.
Espresso is suitable for writing concise and reliable Android UI tests. A test typically contains clicks, text...