Learn Polish: Czy on kupuje nowy filtr i olej raz w miesiącu? - Does he buy a new filter and oil once a month?

To express frequency like once a month, Polish uses the preposition w (in) followed by the Locative case. The dictionary form is miesiąc, but as a masculine noun, it takes the -u ending in the Locative, becoming w miesiącu.

Nowy filtr and olej are the objects he is buying. Why don't their endings change?

They are in the Accusative case, which is used for direct objects. However, for masculine inanimate nouns (objects and things, not people or animals), the Accusative form looks exactly the same as the dictionary form (Nominative case). So, nowy filtr and olej don't change endings here.

Is it okay to say jeden raz instead of just raz?

Yes, but it is usually not necessary. In Polish, raz by itself means once. You would only use jeden raz if you really wanted to stress that it happens exactly one time and no more.

Why use the verb kupuje instead of kupi?

Because the action happens regularly (raz w miesiącu), we must use the imperfective verb kupować (he buys). If we used the perfective verb kupi, it would mean he will complete the action of buying them just once in the future.