Heat 2 tablespoons olive oil in a large nonstick skillet over medium-high heat. Add the potato and cook, tossing, until browned and begaining to soften, 4 to 5 minutes. Add the slice pepper and onion and season with salt. Cook, tossing occasionally, until potato is cooked through and the pepper and onion are browned but not completely soft, 8 to 10 minutes.
Meanwhile, beat the eggs in a bowl and season with salt. Reduce the heat to low and give the pan a moment to cool. Pour in the eggs and cook, stirring constantly, until jet set, 2 to 3 minutes. Remove from the heat.
Toss the arugula with the remaining 2 teaspoons olive oil and the vinegar in a bowl. Season with salt. Fill each roll with the scrambled eggs, cheese, if desired, and the arugula.