In a medium saucepan, combine the barley, 1 ½ cups water, and ½ teaspoon salt and bring to a boil. Reduce heat to medium-low, cover, and simmer until the barley is tender and the water is absorbed, 25 to 30 minutes. Drain and spread on a rimmed baking sheet to cool.
Meanwhile, in a large bowl, whisk together the yogurt, oil, lemon juice, mustard, ½ teaspoon salt, and ¼ teaspoon pepper. Add the celery, apple, mint, and cooled barley and toss to combine.
Divide the arugula among bowls. Top with the barley mixture.