Learn Turkish: Koştuk ve otobüse ancak yetiştik. - We ran and only just caught the bus.

In English, you "catch" a bus as a direct object, but the Turkish verb yetişmek (to reach, to make it in time) requires the dative case. You are literally making it to the bus.

Can I use yakalamak instead of yetişmek for "catching" a bus?

Yes, otobüsü yakaladık is also perfectly natural. Just be careful with your cases: yakalamak (to catch/capture) takes the accusative case (otobüsü), whereas yetişmek (to make it in time) takes the dative (otobüse).

Doesn't ancak usually mean "but" or "however"?

Yes, as a conjunction connecting two clauses, ancak translates to "but" or "however." But when used as an adverb modifying a verb like it is here, ancak means "barely," "only just," or "with difficulty."

Could we use rağmen (despite) from this lesson's topic to express this same idea?

Absolutely! Instead of saying "We ran and...", you could say "Despite running, we barely caught the bus." To do that, you would change the first verb into a verbal noun, add the dative case, and follow it with rağmen: Koşmamıza rağmen otobüse ancak yetiştik.