It's similar to the message appearing in Chrome when a new version is available - something like "Chrome has just got better".
Presuming that you have a server side counterpart, my approach would be the following:
- store the latest version number of your app somewhere on the server (database, configuration file, etc)
- implement an API at server side that, upon an app version provided as parameter, returns true if an update is available
- in your app, read the current app version (see this SO answer)
- call the remote API mentioned in (2), and act accordingly if the return value is true
Note however that appstore notifies the user when an update is available - but that doesn't happen in-app