Without a last name, you might find it down right hard to find out who/whom this person is. There were tens of thousands of soldiers from Louisiana, to include New Orleans that were assigned to the Confederate Army in Northern Virginia. One thing you could do is find out all the Louisiana Units that were in Northern Virginia at the time of the Battle (June 10, 1861) and originated in New Orleans (if this was his home location) then look at their muster rolls to see if you can find someone named Cornelius. This might reduce the number of possible names to a few thousand.
Best of luck