As we all know Apple took Game Center away as an app a few updates back. It can still be used to host SP worlds though & I’m going to show here how!
You and your friend will need an Apple device. Hosting singleplayer worlds is not supported between two Android devices or one iOS and one Android device.
EDIT: Game Center support was removed from The Blockheads in 1.7
Here we go:
This is my singleplayer with custom settings. You need to tap the “options” above the “play” button.
After that you need to tap “host mutliplayer”.
You will have two hosting options. We’ll go with the Game Center one here. Tap “host Game Center…”
Then just “host game”
Something like this should open in a second. Ignore the “Play Now” option and tap “Invite Friends”
iMessage will open. At the top of the screen you can see this. Just tap on it (ignore the +) and start writing the name of your friend.
As soon as it suggests your friend tap on him/her name.
That will send an invitation. At this part you need to wait for your friend to see the invitation and accept it before the game can start.
When your friend accepts the invitation it will say “Accepted” under his/her name.
After a few seconds the world will load and you’re in!
Thanks to @Koppany for letting me use him as an example. Please let me know if some of the information is incorrect.