Learn Portuguese (Portugal): Compraste aquele carro no ano passado?

Compraste is the past tense (pretérito perfeito) for tu (you, informal). Regular -ar verbs like comprar (to buy) drop the -ar and add -aste for the tu form in the past tense. While compras means "you buy", compraste means "you bought" or "did you buy".

Where is the Portuguese word for "did" in this question?

Portuguese doesn't use helper verbs like "do" or "did" to ask questions. You simply use the normal statement—in this case, compraste (you bought)—and add a question mark, raising the pitch of your voice at the end of the sentence.

Why use aquele for "that" car instead of esse?

Both words mean "that", but aquele is used for things that are far away from both the speaker and the listener. Because this car was bought "last year" (no ano passado), it feels distant in time as well, making aquele the most natural choice here. Also, notice it is aquele and not aquela because carro is a masculine word.

Why do we need no in front of ano passado?