Whisk soy sauce and ginger together in a large glass or ceramic bowl. Add steak and toss to evenly coat. Cover the bowl with plastic wrap and marinate in the refrigerator for at least 30 minutes or up to overnight.

Heat a wok over high heat. Add steak mixture; cover wok and cook until steak browned, about 5 minutes. Transfer steak to a plate.

Heat butter in wok over high heat; stir in mushrooms, onion, bell pepper, and sweet and sour sauce. Cover and cook until vegetables begin to soften, about 3 minutes. Stir in cooked steak; cook until heated through, about 2 minutes. Serve over cooked rice.