Soak potatoes in a bowl of water to draw out the bitterness, about 15 minutes.
Rub salt over cucumber and onion slices; squeeze out excess moisture and pat dry. Set aside.
Drain potatoes and place into a pot with carrot. Add enough water to cover. Season with salt. Bring to a boil, then reduce the heat to low and simmer until potatoes can be easily pierced with a bamboo skewer, 10 to 15 minutes. Remove carrot from the pot when tender.
Drain potatoes and return to the pot. Add apple juice and vinegar; mash potatoes with a wooden spatula or fork for about 2 minutes.
Transfer mashed potatoes to a bowl. Add carrot, cucumber, onion, and eggs. Mix in mayonnaise, then season with salt and pepper. Refrigerate until cool, 30 minutes to 1 hour. Serve in individual bowls.