Learn Russian: Мы попали в театр только к восьми.

Дойти до focuses on the physical journey of walking until you reach a specific boundary or point. Попасть в (the perfective past plural form is попали) means 'to manage to get to' or 'to end up in'. Using попали here implies there might have been a delay, traffic, or some difficulty, and they finally managed to arrive, rather than just describing the physical act of walking up to the building.

Why is it к восьми and not до восьми?

К + Dative means 'by' a certain time, acting as a deadline. They managed to arrive by eight o'clock. If you used до восьми, it would mean 'until eight'—for example, 'We stayed at the theatre until eight'. Because they are talking about the point of arrival, к is the correct choice.

What case is восьми in this sentence?

It is in the Dative case, which is required by the preposition к. The nominative form is восемь (eight). When numbers end in a soft sign, they decline similarly to feminine nouns ending in a soft sign (like тетрадь), so восемь becomes восьми in the Genitive, Dative, and Prepositional cases.

Why is в театр in the Accusative case instead of the Prepositional case?