Combine cold water and beans in a pot; let stand 8 hours to overnight.

Stir butter into the pot with the beans and water; bring to a boil, reduce heat to low, cover the pot, and simmer until beans are tender, about 2 hours.

Heat a large skillet over medium-high heat. Cook and stir pork sausage in the hot skillet until browned and crumbly, 5 to 7 minutes; drain and reserve 2 tablespoons sausage drippings.

Stir sausage and reserved drippings, apple slices, tomato juice, onion, brown sugar, garlic, salt, chili powder, mustard, and black pepper into beans. Bring bean mixture to a boil, reduce heat to low, cover the pot, and simmer until flavors blend, about 2 hours.

Stir rum into beans, ladle beans into bowls, and top with sour cream.