You need to run 2 queries - an Append query, and an Update query. Like
Code:
INSERT INTO YourLocalTable (Name, Position, Company, ConnectedOn, MessageDate, Subject, Folder)
SELECT lin.*
FROM
YourLinkedinTable lin (LEFT JOIN YourLocalTable loc ON loc.Name = lin.Name AND loc.Position = lin.Position)
WHERE loc.Name Is Null
UPDATE YourLocalTable
SET MessageDate = lin.MessageDate
FROM
YourLinkedinTable lin (LEFT JOIN YourLocalTable loc ON loc.Name = lin.Name AND loc.Position = lin.Position)
WHERE loc.MessageDate Is Null AND lin.MessageDate Is Not Null