Of course the United Nations has to follow the rule of law. Within their own chambers, they follow their own rule of law. And in the countries they operate in, they follow the rule of law as specified by their hosts. I don't believe the United Nations has ever said they were above it.
The UN is a law making body and, therefore, has to follow laws - after all, they are part of the world and therefore under their own jurisdiction. There can be a debate as to whether they have to follow the UN, but the other way around, there is no question.
The United Nations must follow the Rule of Law in order to consider itself a legitimate governing body representing the countries of the world. Otherwise, the UN is illegitimate and derives its power from nowhere. With that in mind, the Rule of Law is a relatively simple concept that isn't that difficult to follow.
Everyone has to follow the rule of law, and that includes the United Nations. However, things get a little tricky when you're dealing with a supernational organization that is sometimes placed above the law by its host countries. However, thanks to the UN's policy of betterment, I believe the risk of them going rogue is minimal.
Without a doubt, the UN should follow the so-called Rule of Law. The governing body currently doesn't follow this idea, but many people believe it should. Rule of Law ensures that governing bodies are accountable for their actions because they strictly follow these policies without breaking from them at any point.