Learn Russian: Мой брат сейчас живёт в папиной старой квартире. - My brother now lives in dad's old apartment.

Both are completely correct! However, Russian often uses these special possessive adjectives ending in -ин or -ын for close family members (like мамин, бабушкин, папин). In everyday speech, папиной квартире (dad's apartment) sounds a bit more natural and affectionate than the slightly more formal квартире папы (apartment of dad).

What case is папиной in, and why does it have the -ой ending?

It is in the Prepositional case. The preposition в indicates location, so the feminine noun квартире and all its describing words must be Prepositional. While папин has a special mixed declension, its feminine Prepositional ending is -ой, which makes it perfectly match regular adjectives like старой right next to it.

Is there a reason папиной comes before старой in this sentence?

Yes, in Russian, possessive words (like мой, твой, папин) usually come before descriptive adjectives (like старой). Just like in English, saying dad's old apartment sounds right, while swapping the order might accidentally imply that it is the dad who is old, rather than the apartment!

As a quick refresher, how do I know when to use with the Prepositional case like we do here for ?