Just prior to the lunar landing, the command module Columbia is shown emerging from behind the moon far ahead of the lunar module Eagle. At this point, Eagle had dropped into a lower orbit, which increased its speed and placed it well over a hundred miles in front of Columbia. The show is partly correct, in that Mike Collins, in Columbia, was able to communicate with Earth before Armstrong and Aldrin, due to his higher orbit bringing him above the horizon first.
The same CAPCOM (presumably Charles Duke) is seen for both landing and EVA. In real life, there were three different CAPCOMs during this period. During landing the CAPCOM was Charles Duke, post-landing was Owen Garriott, and the EVA CAPCOM was Bruce McCandless.