I think the best precedent is in Acts 17, when Paul was in Athens. According to vv.17-18, Paul debated with Jews, Epicureans, Stoics, and with anybody who happened to be in the marketplace. It should be noted that the Epicureans were atheist (or at least, they rejected belief in any gods who interacted with the world) and materialist. They would have been right at home among the modern scientific atheists: They believed that everything that happens is merely the result of emergent properties of the microscopic particles which constitute everything. They believed in pleasure-based ethics, but not base hedonism as they knew that this would not lead to human flourishing in the long run. They accounted for the apparent fine-tuning of the universe with the multiple universes hypothesis.
We see quite plainly from Acts 17:18 that Paul took the time to reason with Epicureans. The Bible does not record any of the contents of their discussion, so we cannot say how Paul reasoned, but given the similarity of the Epicureans to modern atheists, I imagine the discussions were also probably quite similar to the modern debates. We might take as an example Paul's Mars Hill discourse in vv.22-34, which (from its contents) was evidently not primarily directed at Epicureans or Stoics, but rather the Pagan polytheists: Paul uses something from the Pagan practice in v.23 to launch into a simple description of monotheism in vv.24-25:
The God who made the world and everything in it, being Lord of heaven and earth, does not live in temples made by man, nor is he served by human hands, as though he needed anything, since he himself gives to all mankind life and breath and everything. And he made from one man every nation of mankind to live on all the face of the earth, having determined allotted periods and the boundaries of their dwelling place, that they should seek God, and perhaps feel their way toward him and find him.
A few things to note about the text: Paul is mostly just describing his beliefs to outsiders, but this is not devoid of logical arguments. He makes two broad points that logically follow from God being the creator of all, and both of which draw a sharp contrast with Paganism:
- God doesn't need anything from us, nor does he benefit from our gifts. Since God made the world, obviously if he needed anything he could simply fulfill his own need without our help. We depend on him and not the other way around. That's the logical argument. It also contrasts with Paganism's belief in gods who need us to make sacrifices to them and who feed on us or require our work in some way. I can imagine many people would find the multitude of gods each making distinct demands rather onerous, and these people would certainly perk up their ears at what Paul says here.
- God is the moral arbiter (v.27 - seeking God is the duty that he has allotted). This, again, follows from the fact that he designed us. Obviously, the designer of a thing is the one who determines its purpose. This, too, would be a breath of fresh air for people burdened by Paganism's multifarious gods. The Pagan gods fought amongst themselves and were poor teachers of good. None could claim authority as the arbiter of good. But if one God has made everything, this God can take that role.
Then, in v.28 Paul points to ideas within the Pagan tradition that seem to agree with the monotheistic concept of God. In vv.29-31, he draws his conclusion: If there is a creator God, then we are all accountable to him. In these verses, he brings the discussion back to simply explaining Christianity. He explains God's judgment, the call for repentance, and the resurrection.
In summary: Paul explained Christianity to the crowd, and drew connexions with Pagan tradition to help them understand, as well as to show why Christianity is more satisfying. It is reasonable to suppose that he employed similar tactics in discussion with Epicureans. Add in the fact that the Areopagus was a public debate forum and we have all we need as a precedent for formal intellectual debates with atheists.
