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Adjust oven rack to lower-middle position and heat oven to 325 degrees. Pat pork dry with paper towels and sprinkle with 1 teaspoon salt and ½ teaspoon pepper.
Heat oil in Dutch oven over medium-high heat until just smoking. Add half of pork and cook, stirring occasionally, until brown on all sides, about 8 minutes; transfer to bowl. (Reduce heat if bottom of pot begins to scorch.) Repeat with remaining pork.
Reduce heat to medium. Add onion, bell pepper, celery, tomato, and ½ teaspoon salt to now-empty pot and cook until vegetables are softened and liquid has evaporated, 8 to 10 minutes, scraping up any browned bits.
Add paprika, caraway seeds, and garlic and cook until fragrant, about 1 minute. Stir in water and pork and any accumulated juices and bring to simmer, scraping up any browned bits. Cover, transfer to oven, and cook for 1 hour. Stir in sauerkraut, cover, return pot to oven, and continue to cook until pork is fully tender, about 30 minutes longer.
Using wide spoon, skim off any surface fat. Season with salt and pepper to taste. Serve, garnished with sour cream and dill.