"First Man" was praised for being somewhat realistic in its representation of the events of the Apollo program, and Ryan Gosling's depiction of Armstrong was also praised by the Armstong family for being fairly accurate (even though Gosling seems to say as little in films as he can possibly get away with).

What I don't know is to what extent NASA helped in the technical details of the film. Did the organization help? If so, in what particular areas?

Location, expertise, and consultants.

From the wiki:

with 35mm film being used for the scenes that take place in the Armstrong house and around the NASA facility.

NASA historian Christian Gelzer, as well as astronauts Al Bean and Al Worden, were on set as technical consultants.[24]

From this well-detailed article:

Singer visited NASA headquarters in Washington, D.C. to meet with Ulrich and Chief Historian Bill Barry for advice on the script. “They gave me more books than I can carry,” laughs Singer, who then traveled the country to visit different NASA facilities and interview a number of people who worked either directly with Armstrong or on the missions he served on.

“NASA was incredibly helpful,” says Singer, not only during the writing process (in one case, Singer was able to fly a simulation of Armstrong’s X-15 flight that opens the film in order to write it as realistically as possible, as well as interview the last living X-15 pilot, Joe Engle), but also during filming. “Damien [Chazelle] made an effort to do as much practical shooting as possible,” says Singer. Using NASA’s plans, models—and in some cases, the actual equipment—the crew physically built models of spacecraft, planes, and other machinery for the film.

NASA also was essential in creating the authentic sounds the audience hears throughout the film, providing sound artists with access to Gemini and Apollo spacesuits.

So NASA was on-board and a very large help for the creation of First Man.