In Prometheus, towards the end the Engineer leaves his spaceship to pursue Dr. Shaw, and we last see him on the floor of the escape pod after an alien pops from his chest. In Alien, when examining the spacecraft, the crew finds the Engineer skeleton inside of a cockpit with a hole in it's side. Isn't this supposed to be the same Engineer that chased Shaw? How is he still in the cockpit?
It is not the same engineer. It is not even the same planet.
In 'Alien', the crew of the Nostromo lands on LV -426 where they find the 'derelict' (the wrecked space craft) with the 'space jockey' (engineer) dead in his seat. It is at this point nothing more than a barren planetoid with trace elements of an atmosphere (ASH: "It's almost primordial").
In 'Prometheus', which is set approx. 29 years earlier, the crew of the Prometheus land on LV - 223, a moon with an extremely active atmosphere and dynamic geology.
The events that led up to the crash discovered in 'Alien' have not been covered yet however, after seeing the events of 'Prometheus' one can imagine that the original 'derelict' on LV - 426 encountered similar problems related to the mutagen. Or perhaps David had something to do with it (watching 'Alien: Covenant' answers some questions while raising many more).
TL:DR - 'Prometheus' is not a direct prequal to 'Alien' and they are set on different planets - Alien Universe Timeline.