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You must execute the code again to save those values in the database.

CREATE TABLE dog_walks_monthy_hours (
     dog_walker_id int,
     jan_2010 int,
     feb_2010 int,
  	 mar_2010 int,
     apr_2010 int,
     may_2010 int,
     jun_2010 int,
     jul_2010 int,
     aug_2010 int,
     sep_2010 int,
     oct_2010 int,
     nov_2010 int,
     dec_2010 int
);

INSERT INTO dog_walks_monthy_hours (dog_walker_id, jan_2010, feb_2010, mar_2010, apr_2010, may_2010, jun_2010, jul_2010, aug_2010, sep_2010, oct_2010, nov_2010, dec_2010)
VALUES (1, 160, 160, 160, 160, 160, 160, 160, 160, 160, 160, 140, 160);

INSERT INTO dog_walks_monthy_hours (dog_walker_id, jan_2010, feb_2010, mar_2010, apr_2010, may_2010, jun_2010, jul_2010, aug_2010, sep_2010, oct_2010, nov_2010, dec_2010)
VALUES (2, 80, 80, 80, 80, 80, 80, 70, 80, 65, 80, 80, 70);

INSERT INTO dog_walks_monthy_hours (dog_walker_id, jan_2010, feb_2010, mar_2010, apr_2010, may_2010, jun_2010, jul_2010, aug_2010, sep_2010, oct_2010, nov_2010, dec_2010)
VALUES (3, NULL, 0, 0, 160, 160, 160, 0, 0, 0, 0, 0, NULL);

INSERT INTO dog_walks_monthy_hours (dog_walker_id, jan_2010, feb_2010, mar_2010, apr_2010, may_2010, jun_2010, jul_2010, aug_2010, sep_2010, oct_2010, nov_2010, dec_2010)
VALUES (4, 160, 45, 160, 160, 80, 35, 160, NULL, 80, 40, 0, 0);

WITH pivot_table AS (
SELECT *
    FROM dog_walks_monthy_hours UNPIVOT INCLUDE NULLS (
    hours FOR month IN (
    jan_2010 ,
     feb_2010 ,
  	 mar_2010 ,
     apr_2010 ,
     may_2010 ,
     jun_2010 ,
     jul_2010 ,
     aug_2010 ,
     sep_2010 ,
     oct_2010 ,
     nov_2010 ,
     dec_2010 )))
     SELECT * FROM pivot_table;

DOG_WALKER_ID	MONTH	HOURS
1	JAN_2010	160
1	FEB_2010	160
1	MAR_2010	160
1	APR_2010	160
1	MAY_2010	160
1	JUN_2010	160
1	JUL_2010	160
1	AUG_2010	160
1	SEP_2010	160
1	OCT_2010	160
1	NOV_2010	140
1	DEC_2010	160
2	JAN_2010	80
3	JAN_2010
DOG_WALKER_ID	MONTH	HOURS
4	JAN_2010	160
2	FEB_2010	80
3	FEB_2010	0
4	FEB_2010	45
2	MAR_2010	80
3	MAR_2010	0
4	MAR_2010	160
2	APR_2010	80
3	APR_2010	160
4	APR_2010	160
2	MAY_2010	80
3	MAY_2010	160
4	MAY_2010	80
2	JUN_2010	80
DOG_WALKER_ID	MONTH	HOURS
3	JUN_2010	160
4	JUN_2010	35
2	JUL_2010	70
3	JUL_2010	0
4	JUL_2010	160
2	AUG_2010	80
3	AUG_2010	0
4	AUG_2010
2	SEP_2010	65
3	SEP_2010	0
4	SEP_2010	80
2	OCT_2010	80
3	OCT_2010	0
4	OCT_2010	40
DOG_WALKER_ID	MONTH	HOURS
2	NOV_2010	80
3	NOV_2010	0
4	NOV_2010	0
2	DEC_2010	70
3	DEC_2010
4	DEC_2010	0