Insert Timestamp on value change from other table
Logically (based on the names of your tables), a particular item of evidence cannot be checked out to more than one person at once. We'll assume, then, that no more than one EvidenceTimeStamp table record for a given CaseID and EvidenceNum should have a NULL Evidence_Returned_Timestamp.

In that case, you'd need a trigger on your Outbound Ship Table with something like the following in it:

UPDATE ets
   SET ets.Evidence_Returned_Timestamp = GETDATE()
  FROM EvidenceTimeStamp ets
         INNER JOIN inserted I ON (    ets.CaseID = i.CaseID
                                   AND ets.EvidenceNum = i.EvidenceNum
                                  )
 WHERE i.Outbound_Item_Recevied = 1
   AND ets.Evidence_Returned_Timestamp IS NULL
;


(SQL Server syntax; modify as needed for other platforms)

This will work whether you're modifying/inserting one record or many in the Outbound Ship Table. However, this will update all rows for the given CaseID and EvidenceNum pair(s) that currently have NULL in Evidence_Returned_Timestamp, so if you can have more than one open row for a specific CaseID and EvidenceNum, you'll need some other way to find the right row(s) to update.

(To clarify: If you've got open entries for case 123, evidence numbers 2 and 7, and case 456, evidence numbers 3 and 7, and you updated an Outbound Ship item for case 123, evidence num 7, then the other three open entries are fine.  However, if you had two EvidenceTimeStamp rows with case 123, evidence number 7, then both of them would have their Evidence_Returned_Timestamp set.)

NOTE: Every table should have a column (or group of columns) that is guaranteed to uniquely identify every row in the table and it (or they) really should be the primary key. You may not want to set that up as a clustered index, but you enforce uniqueness by making that the primary key.