A CTE and ROW_NUMBER should do the trick for you:
CREATE TABLE #SAMPLE
  (Cust BIGINT,
  [Date] date,
  [Status] varchar(150),
  Category varchar(50));
GO

INSERT INTO #SAMPLE
VALUES
  (2199996080116, '20160909','Completed with issues', 'UNSAFE'),
  (2199996080116, '20160817','Completed with success', 'NO ACCESS');
GO

WITH MaxDate AS (
  SELECT *,
        ROW_NUMBER() OVER (PARTITION BY Cust ORDER BY [Date] DESC) RN
  FROM #SAMPLE S)
SELECT Cust, [Date], [Status], Category
FROM MaxDate
WHERE RN = 1;
GO

--Clean up
DROP TABLE #SAMPLE;