It's not entirely clear what the relationship is between your tables, or what the conditions are for updating. But something like this should get you started:
UPDATE
N
SET
mid = ML.mid
FROM
nm As N
INNER JOIN mbr As M
ON M.mid = N.mid
CROSS APPLY
(
SELECT TOP 1 mid
FROM mbr As M2
WHERE M2.eid = M.eid
ORDER BY M2.[date] DESC
) As ML
WHERE
N.mid != ML.mid
;
SQL Server CROSS APPLY and OUTER APPLY[
^]