제가 고수를 아니지만 이렇게 하면 될껍니다.

SELECT * FROM
( SELECT
ROW_NUMBER() OVER (ORDER BY 

                                                                  (case
                                                                         WHEN category = '노트북문의' THEN '1'
                                                                         WHEN category = 'AS요청' THEN '2'
                                                                         WHEN category = '기타사항' THEN '3' 
                                                                  END)

                                                                                              ASC) AS ROW_NUM
       :

       :

, case
WHEN category = '노트북문의' THEN '1'
WHEN category = 'AS요청' THEN '2'
WHEN category = '기타사항' THEN '3' 
END AS F_CATEGORY
FROM
_board_computer
WHERE objProperty = ''
) T
WHERE T.ROW_NUM BETWEEN 0 AND 50  ORDER BY T.F_CATEGORY ASC

물론 보긴 별루 않좋지만 이렇게 두번을 사용하시던가 아니면 우선 case 를 사용한 select를 다시 select를 하시면 됩니다.

(CASE category
            WHEN '노트북문의' THEN '1'
            WHEN 'AS요청' THEN '2'
            WHEN '기타사항' THEN '3' 
END)  AS   F_CATEGORY