You need to use pseudo-table "inserted" in the trigger query, otherwise you'll update the whole table. Also, if any row doesn't match the JOIN to OkulKoduSiparisCat , that row won't be updated. You almost certainly need to add the database name on that JOIN.
UPDATE
ORFLINE
SET ORFLINE.SPECODE2 = OkulKoduSiparisCat.Cari_Kodu
FROM LG_020_01_ORFLINE AS ORFLINE
INNER JOIN inserted AS i ON i.?key_col? = ORFLINE.?key_col?
LEFT OUTER JOIN LG_020_01_ORFICHE AS ORFICHE ON ORFLINE.ORDFICHEREF = ORFICHE.LOGICALREF
INNER JOIN MYSQL.?database_name?.dbo.OkulKoduSiparisCat ON OkulKoduSiparisCat.Siparis_NO = ORFICHE.DOCODE