DATEPART(HOUR, [date]) returns the hour in military time ( 00 to 23 )
If you want 1AM, 3PM etc, you need to case it out:
SELECT Run_Time_Hour =
CASE DATEPART(HOUR, R.date_schedule)
WHEN 0 THEN '12AM'
WHEN 1 THEN '1AM'
WHEN 2 THEN '2AM'
WHEN 3 THEN '3AM'
WHEN 4 THEN '4AM'
WHEN 5 THEN '5AM'
WHEN 6 THEN '6AM'
WHEN 7 THEN '7AM'
WHEN 8 THEN '8AM'
WHEN 9 THEN '9AM'
WHEN 10 THEN '10AM'
WHEN 11 THEN '11AM'
WHEN 12 THEN '12PM'
ELSE CONVERT(varchar, DATEPART(HOUR, R.date_schedule)-12) + 'PM'
END
FROM
dbo.ARCHIVE_RUN_SCHEDULE R