When you enter a sporting event, or join a sporting league, or whatever, that is organised by the GAA, or the IAAF, or some such body, there's a contract between you and the organisers, one of the terms of which can be that both parties agree to submit disputes arising out of the the contract to binding arbitration. Similar terms are often included in, e.g. major construction contracts, and a variety of other types of contracts.
They generally can't be included in consumer contracts — using the small print to require consumers to surrender their right of access to the courts is generally seen as contrary to the unfair contract terms legislation that applies to most consumer contracts. So you can still bring your dispute about the dodgy toaster you bought, or the flight that you were bumped from, to the District Court small claims procedure.
Most defamations don't arise in the context of any contract at all, so a contractual pre-agreement to submit a defamation claim to arbitration isn't often a possiblity.
That's not to say that defamation claims can't go to arbitration — they can, but in most cases the agreement to submit them to arbitration arises after the alleged defamation has happened. The one exception that I know of is that newspapers that participate in the UK Independent Press Standards Organsation have a standing contractual obligation (to the IPSO) to accept the jurisdiction of IPSO's arbitration scheme, if a plaintiff wants to invoke it. But plaintiffs don't have to invoke it; they can go to court if they prefer.
One of the main attractions of arbitration is that proceedings are conducted in private. So the plaintiff can bring his claim without having the alleged defamation repeated and republished and brought to the attention of a wider and wider circle of people in the way that it would be in court proceedings. On the other side, the defendant benefits because, if they lose the action, they won't suffer reputational harm by becoming publicly known as the publisher of defamatory material — this matters if you're, e.g, a newspaper or a journalist with a reputation to guard.
The main downside to arbitration in defamation cases is that if what the plaintiff wants is public vindication of his character, he won't get it in arbitration proceedings, because nobody will get to hear about them. The other main downside is that somebody has to pay the arbitrator. A typical fee for professional arbitration is anywhere between €7,500 and €15,000, which is normally shared equally between the parties. That's a lot more that the court fees you have to pay to issue Circuit Court proceedings.