Your package will have two Connection managers. In this case, I'll assume OLEDB but ADO.NET or ODBC will also work and further assume they point to Server1 and Server2 and are named as such.
The pattern you are going to have is two Execute SQL Tasks in serial. The first Execute SQL Task will ask Server1 what the value of EXEC_END_TIME is and store that to an SSIS variable.
Create a variable named LastExec and set the data type as datetime and initialize it to something like 1900-01-01 at midnight
In the Execute SQL Task, change the result type from None to Single Row and then on the Result Set tab, map the 0th element to the variable
See also How to set a variable using a scalar-valued tSQL function in SSIS SQL Task
The second Execute SQL Task will update statement as Panagiotis describes and the "magic" will be using the SSIS variable in the query.
UPDATE ThatTable SET ThatField=?
The ?
is the place holder for OLEDB and ODBC connection manager queries. The difference being OLE is 0 based ordinal and ODBC is 1 based. ADO.NET will used named parameters so the original comment query would work.
In the Parameter Mapping tab, you will need to associate the SSIS variable with ordinal position 0 of your query. Sample screen Logging information when executing remotely SSIS from ASP.NET
Get the first execute sql working first. Once you can query the database and assign to a variable, getting the next one (set as a successor) in line should be a snap.
Data flow approach
You could continue with your data flow approach. Instead of an OLE DB Destination, you'll use an OLE DB Command. Use the same query and this time, you'd map the source column to the zero-eth element.
It's overkill so that's the reason I did not advocate for its approach.