Qualifications
Arbitrators are often unilateral decision makers who leave little room for appeals. In other words, they'd better know what the heck they're doing. The more "expert" these experts can be, the better, leading most of them down a rigorous career path spanning college, work experience, and training before even being considered for the role of arbitrator.
First up, there's school. If you want to arbitrate a very narrow scope of cases, you can sometimes get away with a bachelor's degree only (especially if it's related to that narrow field). But since you need one no matter what you want to arbitrate, a bachelor's is a great place to start regardless.
Arbitrators with big ambition (and, eventually, big student loan payments) should follow up on that degree with some post-graduate education. This could be a master's in dispute resolution or a full-on doctorate (source).
Next up, training. You'll need to pass the bar exam in whatever state you're planning to work in, which means about a million hours of studying (that's our ballpark estimate). After that you'll need special training sessions in mediation and arbitration. In some states, you'll be required to have your CPA as well, so tack on another million hours of studying while you work toward that.
Finally, there's the experience you have to build up. Sort of feels like you'll never be an arbitrator at this rate, doesn't it? You can have less experience if you want to work privately and on something super specific, but court-appointed arbitrators are typically asked to show about ten full years of relevant experience to even be considered for an official court roster (source).