By some misfortune and bad luck I ended with Schengen visa of 8 stay days and the flight back is one day later. So, I shall have one day overstay. Does it have any sense to show up in the airport one day early, so that I stay within stay day limits?
In order to avoid overstaying you'll have to clear border control exit checks before 23:59:59 on the last visa validity day. It might make sense to arrive (very) early in the airport in order to make the cutoff time. However the problem is that border control checkpoints are usually located after airport security checks. So the question really becomes: how early before my flight can I go through airport security checks?
The access to security checks is likely to be regulated by flight departure time. So is check-in, although you can always do that online and be alright if you don't have any hold-luggage to drop off. Keep in mind that you'll need a boarding pass to be allowed through security. I couldn't find any definite answer on the topic, and some forum posts seem to say that this is airport dependent. Personally I would call the airport and ask, not mentioning the overstaying issue but rather saying for example that you have an early departure and don't want to sleep in a hotel. I would also turn up at the airport and try my luck anyway.
As a final note make sure you make it through security checks at a reasonable time, i.e. one that is compatible with the opening hours of border controls. Indeed these might not be manned 24h, so you want to make sure that you arrive at the right time.