Finally saw this over the weekend. Who knew the story of the first man to set foot on the moon could be so deadly boring. I know what the filmmakers were going for, but they missed, and the movie just wasn't very engaging. I know Neil wasn't the most dynamic personality, but Gosling's portrayal made him seem more just out to lunch than anything else.
Can't comment meaningfully on the technical aspects. Some things looked wrong (such as the X-15 being below the clouds when they're still waiting to gain aerodynamic control), but I chalk that up to maybe mistakes in editing. ShakyCamTM got annoying in a number of places.