He probably knows he screwed up majorly, has to call upon arrival etc - but still has to do a 8h or so flight. Is he mentally fit to do that? Are there any rules for that?
Just to make a comparison with another transportation industry which has to take safety seriously, consider what would have happened on a railway:
Assuming that the Egyptair flight had passed a location where it was supposed to stop, or took a wrong turn resulting in a runway incursion (not a rush to judgement, but an assumption for the purpose of the example) it would mean the aircraft had moved beyond where it was authorized by the ATC.
If a train crew made a similar mistake, like if a train passed a signal telling it to stop without stopping, or it entered a zone where it had been told maintenance forces were active without first contacting the maintenance supervisor by radio, it would be considered a major violation of its operating authority.
Assuming the violation doesn't result in an accident, once the violation was noticed, the train would be immediately brought to a stop, and the entire train crew would be relieved from duty, whether on a passenger train or freight train. Other than perhaps moving the train to a location where it wouldn't block road crossings or other train movements, the train wouldn't move again until a new crew could be brought to the location.
The rail traffic controller might also be relieved of duty, if the violation involved the controller's instructions or actions, or lack thereof.
Any employees taken out of service would then likely be subjected to an immediate drug test and an investigation. They would then have to endure any penalties that might be imposed following the investication before they would be allowed to return to duty.
The reasoning on the railway's part is that they don't know why the violation occurred, and whether one of the crew had a medical problem, or was under the influence of something. Until they can be sure of the cause, they can't afford to allow the crew to remain on duty.
One major difference, of course, is that trains can be brought to a stop pretty well anywhere to sort things out, which is not an option for an aircraft at FL 370, but it is still interesting that in an industry that tends to be far stricter than railways in the interpretation and enforcement of rules would allow a crew that has been involved in a major violation on the ground to continue, whether at fault or not.
As to crews that might not have broken any rules, but were involved in a close call, such as if the train they were operating struck a car at a road crossing, or they were on a train that could have been in an accident because of the rule violation of another train, the members of the crew are given the option to be relieved by a fresh crew if they want it. Note that the pay system does not penalize them if they ask for relief, other than they don't get paid again until they come back to work, or a doctor mandates some time off to recover.