Assumption: ISS has its typical crew of 3 astronauts and 3 cosmonauts with two Soyuz modules docked at the time. There is a free docking port. Further assumption: the early announcements were published immediately, but not directly to the newspapers. Certain astronomers would know very fast in a rather uncontrolled manner though.

Hawaii Space Telescope, T - 3.25 days:

"We are tracking a previously undiscovered near-earth crossing asteroid."

Hawaii Space Telescope, T - 2.5 days:

"We are tracking an earth-crossing asteroid in a high-hyperbolic solar orbit. It will pass 150km over the surface of the earth over Egypt in 1.5 days. Spectroscope indicates asteroid surface is nearly pure titanium." There is a brief twitter storm discussing capturing the asteroid, but this is obviously infeasible.

Hawaii Space Telescope to NASA HQ in Houston, Classified, T-1.5 days:

"Asteroid has asymmetrical repeat asymmetrical strong thermal signal. Looks like a course correction burn."

Hawaii Space Telescope, T - 1.4 days:

"Earth-crossing asteroid will pass 35km over Greece in 10 hours. It will be readily visible." This makes the news broadcasts. There would be no point suppressing this anyway as everybody on that side of the planet would be able to see it.

Space, T - 1.3 days

Spacecraft crosses Moon's orbit. Crew is locking down for aerobrake maneuver. They have shuttered their radio equipment, and so can't hear anything even if we tried.

NASA HQ Houston to Pentagon, Top Secret, T - 1.3 days:

"Confirmed unknown spacecraft attempting aerobrake maneuver over Greece in 8 hours. They are not transmitting at this time."

T - 1.25 days:

Secret Kremlin briefing involving dispatch from Hawaii. (They weren't set up to send classified messages well enough.) US knows of the briefing but not the contents. It doesn't raise immediate alarm because this happens every few days.

Pentagon to President, Top Secret, T - 1.2 days:

"In 5 hours a previously unknown spacecraft launched by an unknown power will attempt to enter earth orbit or possibly land in Europe. Spacecraft is controlled, but not by us. The attempt will be visible to all of the middle east, northern Africa, all of Europe, and parts of western Asia including important Russian cities."

Europe, T - 1 day:

Thanks to the news announcement, far more people than normal have turned out to watch the asteroid cross the sky above them. This spectacular high-speed event would be well-remembered and well-photographed as few people expect that anything like this would happen ever again.

Near Earth space, T - 16 hours:

Spacecraft conducts orbital insertion burn with correct phase adjust to rendezvous with ISS. (I'm assuming they got lucky and could intercept after only 1 orbit.) This event is observed by multiple NASA, Russian, and Australian telescopes.

Near Earth space, T - 15.5 hours:

Spacecraft radio deployed. They are now listening for any attempt at radio contact from us. They don't speak English or any other Earth language, and their FM isn't compatible with ours, but their AM is.

NORAD to Pentagon, White House, Top Secret, T - 15 hours:

"Spacecraft is attempting to rendezvous with ISS. They obviously know we're here. We've quietly gone to Defcon 4 just to bring all the monitoring stations to full staffing. Somebody tell the space station they're coming."

Low earth orbit, T - 8 hours:

Spacecraft makes second maneuvering burn for fine-grained intercept of ISS in 8 hours.

Low earth orbit, T - 0 hours:

Spacecraft is now holding 300 meters from ISS. If they haven't been contacted yet, they try to radio the ISS on their own frequencies. We can obviously understand them no more than they could understand us, but the message is unmistakable anyway. "We are here!"

What goes on in space I know what to do with. The reactions of the people observing this on the ground defeats me. The kind of immediate thing in the next few hours in politics as the paradigm shifts to we are not alone is lost on me. The question is best answered by injecting points into the timeline in between T-1 days and T+ a few hours. It really is more about politics than anything else.

You see, I know who the aliens are, and I know what must unfold over the next several days, but for these few hours, the people of the world are in the dark. They know they are not alone, but haven't a clue what has found them. I know step by step the correct policy from knowledge, but they have none and I cannot guess. My weakness, I cannot fathom the response of ignorance combined with politicians fallacy. Do nothing is the right move, but they surely won't do nothing.

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    $\begingroup$ I'm not entirely sure what question you are asking here. If you are asking about what happens next, that is something WorldBuilding.SE calls "idea generation" because there really is no right or wrong answer, it's all personal opinion. If you're asking what happened in the question, it sounds like you already know. Personally, I can't think of very many people I would rather represent my species than our astronauts, so I'd be happy with the events. $\endgroup$ – Cort Ammon Mar 26 '16 at 22:37
  • $\begingroup$ When do the people find out that it is a spacecraft, not an asteroid? $\endgroup$ – CHEESE Mar 27 '16 at 12:46
  • $\begingroup$ @CHEESE: Probably, at T - 8 hours, unless somebody leaked it earlier. The politics of the matter defeats me. $\endgroup$ – Joshua Mar 27 '16 at 15:43
  • $\begingroup$ Related: Is there no physical security in space, other than being in space? on Space Exploration. $\endgroup$ – a CVn Mar 31 '16 at 7:48