Perhaps the way to think about it is to ask the question, Why would he need to be mentioned?
What happened during his reign, 37-41AD, that would have gotten him a mention in Acts?, the only thing I can think of that would have gotten his name in the book would have been if Luke would have referenced him in explaining how Herod Agrippa came to power, but Luke saw no need to do so, and by the time Agrippa is mentioned, Caius is already pushing up daisies.
Another point would be that Christianity at the time of Caius's reign wasn't even a blip on the Roman radar screen yet, so he had no specific interest in it, he may have unknowingly been used in the rule of God over the affairs of men to help the spread of the Gospel because he did distract the Jews for a time by insisting his statue be placed in the Temple which probably took their minds off those pesky followers of Jesus.
Quite a few folks don't get a mention in the Gospels or Acts that you might think should, what about the procurators before and after Pilate, what about the high priests after Caiaphas, what about Nero?
I enjoy studying the history of that period, I believe in the Providence of God, that He is the shaper of history and that His hand prepared the world for the spread of the Gospel, and I try to understand how the world looked, how the people lived, how events affected them and how God works all things together for good to those who are called according to His purpose.
Have your read much Josephus or Tacitus?