You can use this approach by using variable to store data and then use it foe next column.
DECLARE @val VARCHAR(20) -- type of Col1
UPDATE THINGS
SET @val = [Col 1] = CASE WHEN 'A' THEN 'APPLE'
WHEN 'B' THEN 'BALL'
WHEN 'C' THEN 'CARROT'
END,
[Col 2] = CASE @val WHEN 'APPLE' THEN 'FRUIT'
WHEN 'BALL' THEN 'TOY'
WHEN 'CARROT' THEN 'SALAD'
END