Okay here we go. The episode was Tomorrow is Yesterday.
Clue 1: An unplanned visitor.
....The first unplanned visitor was Captain Christopher.
Clue 2: Talk about lucky timing.
...It's a time travel episode where an accidental sling-shot not only sends them back in time, but they end up at Earth over the United States in the 1960s. Lucky. Also, the show gets the moon landing happening on a Wednesday part right.
Clue 3: So close to the actual term.
...The catalyst for the episode's circumstances is "black star of high gravitational attraction." The term "black hole" won't be popularized until the early 1970s.
Clue 4: "Hey! I thought Star Trek was on...oh"
...The opening teaser features nothing to indicate you are watching a Star Trek episode until the Enterprise appears in the last few seconds.
Clue 5: Oops! Another unplanned visitor.
...The Air Sergeant is beamed aboard after activating the emergency signal.
Clue 6: It's good for the soul. <------I really thought this one would give it away.
...Chicken soup
Final clue: By that statement, did Kirk mean twelve or thirteen?
...The age old question on the number of Starships like the Enterprise.
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There you have it. I pass the baton to Discofan.