Learn Finnish: Metron pysähtyessä kuulemme tärkeän kuulutuksen. - When the subway stops, we hear an important announcement.

This is the core of the temporal construction. Metron is in the genitive case and acts as the subject of the action. Pysähtyessä is the second infinitive inessive form (ending in -essä) of the verb pysähtyä (to stop). Together, they translate to "when the subway stops".

Could I just use the word kun (when) instead?

Yes! You could say Kun metro pysähtyy, kuulemme... and it means exactly the same thing. The temporal construction is just a more compact, slightly more formal way to express two things happening at the same time.

Why do the words tärkeän kuulutuksen end in -n?

They form the total object of the sentence. Because we hear the entire announcement (a completed action) and the main verb kuulemme is a standard active verb, the object takes the accusative case that looks identical to the genitive case, ending in -n.

The word kuulutuksen looks related to the verb kuulla (to hear). Is it?

Exactly! Kuulla means "to hear". From that, we get the verb kuuluttaa, which means "to announce" (literally to make someone hear). Finally, adding the -us suffix turns it into the noun kuulutus (an announcement), which here takes the -n ending for the object.