Sorry for the long awaited update. Apparently there was something wrong with my old world, maybe because I already bred them, I don't know. I've made an achievement world in which you can get basically all achievements.
You first need to kill the ender dragon and then enter the portal, to get back to the overworld. Then, head back to the end of the alley, and then go to the area on the right which says update 0.15. In here, you should be able to find a chest with both horse and donkey spawn eggs, spawn both of them, tame both of them by keep trying to ride them.
Taming the horse should get you another achievement.
Then, give both of them a golden carrot, to unlock your achievement.
I've copied my world and placed this folder on dropbox. You just need to unzip it before you can use it. In this world there's a donkey and horse right where you spawn. You might have to wait a little before you can breed them again, though they are tamed and you don't need to go through all the trouble of finding them and getting golden carrots. Download the "artificial selection" world.Achievement worlds
You first need to tame a donkey and a horse. Then feed them a golden carrot. If you're using my world, wait a while before you can feed them again, since they are already fed and tamed.