What should someone do if he has fats to make up but didn't finish before next Ramadan. Should he fast them after the next Ramadan, pay kaffarah/fidyah (idk which one) or just pay a compensation?

He fasts them after the next Ramadan, as they remain a debt on him. If the delay was due to an excuse such as illness, nothing else is due. If it was without an excuse, he is sinful and must repent, and the scholars differ: Shaykh Ibn Baz held he must also feed one poor person for each day, and Shaykh Ibn Uthaymin held only the fasting is required.