Oh my, this could be quite a long answer, so in order to keep it to a manageable length I'm going to concentrate on one author, Justin Martyr in the 2nd century. Naturally, a lot of anti-pagan apologetic is pre-Constantinian, though it remained an issue well past the legalization of Christianity in the early 4th century and even past the declaration of Christianity as the official state religion in the late 4th century; even Augustine of Hippo, straddling the 4th and 5th centuries, still had to deal with paganism in North Africa, so it was a live issue for the Church for at least a good four centuries.

Anyway, to Justin Martyr. He wrote two Apologies, one addressed to the Roman Emperor Antoninus Pius, the second to the Roman Senate. He relied largely on an emphasis on reason, and the identity of Christ as Logos from John 1 - usually translated "Word" but having a lot more meaning than that in Greek philosophy, more like "reason" or "rationality". His argument was that Christianity was more logical and rational than Classical paganism, and in fact appealed to Socrates and his execution for "introducing the worship of new gods" as an example of the rational worship being persecuted by pagans.

He goes on to defend Christians as good, law-abiding, untroublesome citizens who deserve no opposition or persecution, who live lives of chastity, charity, and peace. He also defends the historicity of Christ's resurrection and gives a long catalogue of Old Testament prophecies fulfilled by Christ.

Particularly interesting to me is how he attributes to Plato knowledge of the Pentateuch:

And the physiological discussion concerning the Son of God in the Timæus of Plato, where he says, He placed him crosswise in the universe, he borrowed in like manner from Moses; for in the writings of Moses it is related how at that time, when the Israelites went out of Egypt and were in the wilderness, they fell in with poisonous beasts, both vipers and asps, and every kind of serpent, which slew the people; and that Moses, by the inspiration and influence of God, took brass, and made it into the figure of a cross, and set it in the holy tabernacle, and said to the people, "If you look to this figure, and believe, you shall be saved thereby." And when this was done, it is recorded that the serpents died, and it is handed down that the people thus escaped death. Which things Plato reading, and not accurately understanding, and not apprehending that it was the figure of the cross, but taking it to be a placing crosswise, he said that the power next to the first God was placed crosswise in the universe.

Justin Martyr, First Apology, chapter 60. His appeal to the rationality of the Christian faith is therefore linked with an affirmation of the best of Greek philosophy, which likewise rejected the pagan pantheon, and is a sign of some future theologians who see great wisdom and commonality in the Greek philosophy.