I've been referred to posts such as this in favor of the argument, but it's pretty thin on sources and sounds a little hagiographic. It's missing the discussion of women's rights, ethnic tensions or homogeneity, resource conflicts, or any of the troublesome issues that an anarchist society of the day would have to deal with. What was up in Ireland before Cromwell? It what ways was it an anarchist system of government? How was this system tested? How were disagreements settled?
I would definitely disagree with this article. It is pure fantasy. It seems to use a strange version of a stipulative definition under which Ireland was anarchist because it didn't have a modern nation state. It did have kings, laws, private property, slaves and unequal gender relations so I don't know what kind of anarchism that is. Part of the problem is that we lack sources for much of the period before 700 AD so it is easy to just make stuff up.
Without talking about irish specificities, it's a pretty obvious ideological argument to call one society anarchist. It isn't bad in itself, but one must be aware of it. Of course, any ideology must refer to the past in some way to position itself and it's programmatic dimension. Any utopian project (or plain alternative political wordview) such as anarchy will want to point out pre-modern societies to explain how they worked without what we undestand today as a State or as a Governement. It is a kind of allegory that fleshes out a few caracteristics of today's social distribution of power while showing that alternatives did exist in the shape of complex societies without monopoly of power or state apparatus. Ireland only exists as a comparision in the allegory. I guess about any issue you might bring up will show that Ireland wasn't the perfect model anarchist society, or simply that it didn't describe itself as such. Do not forget that the propriety of an utopia is specifically that it doesn't exist in the historic, contingent real world.