Whisk together vinegar, yogurt, mustard, and the walnuts in a small bowl; season with salt and pepper. Add walnut oil in a slow, steady stream, whisking until emulsified.
Cut potatoes crosswise into 1/2" thick rounds. Bring a medium saucepan of salted water to a boil. Add potatoes, and cook until tender, about 10 minutes.
Using a slotted spoon, transfer potatoes to a colander to drain. In a bowl, toss the potatoes with the dressing.
Return pan of water to a boil.
Prepare an ice-water bath. Add green beans to boiling water and cook until tender and bright green, about 4 minutes.
Using a slotted spoon, transfer to ice water to stop the cooking. Drain and add to the bowl with the potatoes. Toss to coat beans with the dressing.
4. Add arugula to potatoes, and green beans. Toss to coat. Chill until ready to serve.