The trailer shows Gemini scenes. And indeed Armstrong's performance during that mission, when a roll thruster got hung up firing and they rolled so fast they nearly blacked out (would have thus died) before solving the problem, certainly helped to move him up thru the ranks of many other astronauts to be among the select few to be considered to be in a major role for a lunar mission.
The movie also has some X-15 scenes.
And definitely an LLRV scene. Another case of Armstrong being cool, recognizing problems and making the right decisions in a life or death moment.
I do NOT expect the movie to try to follow his whole life. Probably a little bit as a kid, then move quickly onwards, to learning fly, then getting more focused by the time of the X-15 flights (also meeting his wife, getting married and having kids). After showing scenes of Apollo-11's return, the movie will probably be over within 10 minutes, rather than spend much time on him and his family in the decades afterwards.
Screenshot from trailer: