In a medium saucepan, combine the barley, 1 1/2 cups water, and 1/2 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, about 30 minutes. Drain, spread on a rimmed baking sheet or tray, and let cool.
Meanwhile, in a large bowl, whisk together the yogurt, oil, lemon juice, mustard, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Add the celery, apple, cooled barley, and mint (if using). Toss to combine.
Divide the arugula among bowls. Top with the barley salad.