So we have two steps, like "come to the airport (from home/hotel)" and "come to the destination airport (from the flight)." Can we use erreichen (reach) in both cases?
Here are two examples:
We will soon reach our destination airport.
Wir werden bald unseren Zielflughafen erreichen.
Half an hour after leaving our hotel we reached the airport.
Eine halbe Stunde nach dem Verlassen unseres Hotels erreichten wir den Flughafen.
So the short answer is yes you can use "Flughafen erreichen" in both cases.
I don't know on which level you are with your German studies, but I hope that my examples were not too complicated.
Yes, erreichen applies to both situations, and you will be understood. That said,
when you talk about arrival at an airport to pick someone up, you could simply say, for example,
Wir waren gestern am Flughafen...
If you go there by car, train, bus, etc. to get on a plane, this would sound natural:
Ich bin schon um 03:00 am Flughafen gewesen, und hatte genügend Zeit.
If you opt for kommen / ankommen instead:
Sie ist mit dem Taxi zum Flughafen gekommen.
Der Flug war pünktlich, er ist vorhin in Zürich angekommen.
If you are on a plane, landing, we might say,
Bald werden wir in Berlin landen
As so often, native speakers have many ways of expressing this, avoiding the textbook example most of the time. While it is not wrong at all to use erreichen in both contexts as you suggest, it is something I'd expect to see in formal writing or public announcements as by a cabin crew, less so in a conversation.
In regard to public transport, for some reasons erreichen seems to be more popular with trains:
On an ICE:
Wir erreichen in Kürze Frankfurt Hauptbahnhof. Ihre nächsen Anschlüsse: ...
On a Eurowings flight:
Wir landen in Kürze in Graz. Das Wetter...