It's true but it's also misleading. For instance, it's true that Johnson didn't have to run across the campus to the bathroom, but Jackson did. (And someone did complain about Johnson's use of the unmarked bathroom, but only after she had been working there for long enough she had the social capital to ignore the complaint until segregation nominally ended.)

The film does suffer from trying to compress a whole office of computers into just three people, and a whole remaining government bureaucracy into another handful (Costner, Parsons, etc). It also has to do this while covering a part of the space program most people today are not familiar with (technically NACA not NASA, Atlas and earlier). At the same time, it's reasonable to cut out the 90% that's normal office work, because it's boring and doesn't help us understand the time or situation.

Most of the events have strong factual footing, with the notable exception of Costner's white savior moment. If you want more information, the book is dry and a little disorganized, but of course much more thorough about the timeline and who did what.