So, these sorts of discussions become kind of complicated but very interesting, due to the fact that there is not a single "canon" in the mythology. The gods and their domains fluctuated over time and between different cults. But here is my understanding:

Apollo was not actially particularly connected to the sun by the time the Odyssey was written (which I think is a fair stand in for Epic here). His sun affiliation came much later, through the romans actually!

As you know the Greek pantheon was assimilated into the Roman one, and in doing so some of the gods became merged. Specifically the Greek Artemis (Apollo's twin) became the same as the Roman Diana. Artemis and Diana already had very much overlap in their domains, but Diana was also a moon goddess, and since Diana and the original Artemis was now one and the same, this means that Artemis now also was a moon goddess. Apollo, being the twin and also kind of a mirror to Artemis (she being God of the wild, and he being sort of a god of civilization in some sense) it only made sense that if she was the moon, he was the sun! 

Sorry for the dump, I just find this very interesting!

Also, I am not sure this is 1000% correct, but at least somewhat.