At one point he is seen catching a floating pen that is suspended in zero-gravity as he flies a plane to the outer reaches of the Earth's atmosphere.
Gosling and Foy are also captured in scenes in the Armstrong household which see them comforting their on-screen children as they voice their fears about their father taking part in the dangerous mission.
Captions in the clip invite viewers to "experience the improbable journey of the most dangerous mission in history".
The final scenes of the clip tease the moments of the famous landing and space walk. Brief shots see the spacecraft hurtling towards the surface of the Moon.
Momentary glimpses are provided of the astronauts' view of the planet and of a close-up shot of one of the crew climbing down from the spacecraft and on to the lunar surface.
The drama adventure film has been directed by Damien Chazelle, who won the Academy Award for Best Director in 2017 for musical romantic comedy La La Land, which also starred Gosling.
Schindler's List and Saving Private Ryan director Steven Spielberg has an executive producer credit for First Man.
First Man arrives in New Zealand cinemas on October 11.