Is observing prayer time prioritized over purity, or is purity prioritized for someone who woke up in a state of ritual impurity, such that if he were to perform ghusl, the sun would rise? Should he perform ghusl and make up the prayer, or should he pray before sunrise?

He should perform ghusl (definition) and pray, even if the sun has risen, due to the generality of Allah's Almighty saying: (O you who have believed, when you rise to [perform] prayer, wash your faces and your forearms to the elbows) and due to the Prophet's (peace and blessings of Allah be upon him) saying: (Whoever sleeps through a prayer or forgets it, let him pray it when he remembers it).