Why is there no Realms button on my Pocket Edition 1.13?
From your screenshot I can see that you are playing on a beta release of Minecraft. Unfortunately, according to this Minecraft support article on realm connectivity issues, realms are only available when you are playing on the latest public release version of the game.
Make sure you're not using a beta version of Minecraft. If you are, leave it and restart the game (Realms only works with release versions)