How come Apollo, a Greek god of sun, has no Roman counterpart as one would to expect when it comes to Roman mythology? In other words, what was the name of a Roman god of sun?

The Romans changed the Greek gods' names to their Latin equivalents. With that, the Greek Phoebus Apollo (Also simply known as "Apollo") remained as Apollo for the Romans.

Also, Apollo was the god of many things (such as light, prophecy and medicine), not just the sun. If you're looking for the god who is recognized specifically as "God of the Sun", that would be the Roman counterpart of Greece's Helios - Sol.

Well, just as you said, the Greek gods have their Roman counterparts, because the Romans pretty much adopted their entire religion. The only changes they made were the names. (Plus they added one or two gods of their own inventions. Most notably Janus, the two-faced god.) Among the most important Greek(and Roman) deities Apollo is indeed the only one whose name the Romans didn't change. So basically, answering your question, Apollo's Roman name was Apollo. That's it. There is no catch.
By the way Roman poets often only refer to Apollo as Phoebus , but that's sort of just a descriptive term (it means "bright") and the Greeks were also using it, calling him Phoebus Apollo.
Anyway I hope this answers your question. :)

The Romans called him Apollo Phoebus, but most of the times they just called him Phoebus for short. There's your answer. Phoebus.