Stir together the apple, celery root, shallot, vinegar, tarragon, mustard, and 2 tablespoons of olive oil in a medium bowl. Season with salt and pepper, to taste. Let marinate at room temperature for 30 minutes.
Preheat the oven to 400°F. Heat the remaining 2 tablespoons of olive oil in a large ovenproof skillet over medium-high heat. Season the pork on both sides with salt and pepper and add to the skillet. Sear for 2 to 3 minutes per side, until a golden brown crust forms. Transfer the pan to the oven and cook for 12 to 15 minutes, until a meat thermometer inserted into the thickest part of the pork registers 160°F. Remove from the oven and transfer the pork to a cutting board. Tent with aluminum foil and let rest for 10 minutes.
Slice the pork into ½-inch thick slices and arrange on plates. Stir the arugula into the celery root slaw and spoon over the pork.