Learn Finnish: Kissan omistaja ostaa paljon ruokaa. - The cat's owner buys a lot of food.

This is the genitive case, which shows possession or ownership, just like the apostrophe-s ('s) in English. Kissa is cat, so kissan means cat's.

Does the possessor always come before the thing it owns?

Yes! In Finnish, the word in the genitive case (the owner) almost always comes right before the thing being owned. So it is always kissan omistaja (the cat's owner), never the other way around.

Why does ruoka have an extra a at the end to make ruokaa?

This is the partitive case, and it is used here because of the word paljon (a lot). In Finnish, words expressing a quantity or amount always force the noun that comes after them into the partitive case.

Doesn't ostaa also mean 'to buy'? How can we tell it means 'buys' here?

You are exactly right! For this specific group of verbs, the dictionary form (to buy) and the third-person singular form (he/she/it buys) look exactly the same. You can tell it means buys here because it is acting as the main action for the subject of the sentence, kissan omistaja.