Acts 5:33-39 gives an account of speech by the first century Pharisee Gamaliel, in which he refers to two movements other than the Way. One lead by Theudas (v 36) and after him led by Judas the Galilean. Josephus placed Judas about 6 AD. He places Theudas under the procurator Fadus 44-46 AD. Two problems emerge. First, the order of Judas and Theudas is reversed in Acts 5. Second, Theudas's movement comes after the time when Gamaliel is speaking.
First, let me make a general observation. Consider that if we have to documents of historical writing, document A and document B, and they both record events X, Y, and Z. With regard to X and Y they are in perfect agreement, by in regard to Z they differ in their accounts. So obviously document A is wrong, correct? No. That would be an unwarranted assumption. All things being equal, it might also be the case that document B is wrong or it may be the case that document A and B are both wrong. Or it may even be the case that both A and B are correct and their apparent discrepancy is due to some data that we are missing. Perhaps there is some third document, C, that provides the missing data.
My point is that if Josephus contradicts the Bible on some historical point, this shouldn't in itself give us some sort of epistemic angst in regards to the accuracy of the Bible. Why would we assume that Josephus got it right rather than Luke? And in fact we, as Christians, have excellent reasons for thinking that Luke got it right: his writing of Acts was God-breathed. So we shouldn't immediately think something might be wrong with Luke (or any other book of the Bible), but that something might be wrong with Josephus (or any other historian) or that we simply don't have all the information.
Now, in regards to the particular discrepancy you mention. The Theudas you're referring to is spoken of in Antiquities 20.96. Looking at my edition (trans. Whiston) there is a note there that says "This Theudas, who arose under Fadus the procurator, about A.D. 45 or 46, could not be that Theudas who arose in the days of the taxing, under Cyrenius; or about A.D. 7, Acts 5:36–37. Who that earlier Theudas was, see the note on 17.10.5."
The note on 17.10.5 reads "Unless this Judas, the son of Ezekias, be the same with that Theudas mentioned Acts 5:36, Josephus must have omitted him; for that other Theudas, whom he afterwards mentions under Fadus, the Roman governor, 20.5.1, is much too late to correspond to him that is mentioned in the Acts. The names Theudas, Thaddeus, and Judas differ but little. See Archbishop Usher’s Annals, at A.M. 4001. However, since Josephus does not pretend to reckon up the heads of all those ten thousand disorders in Judea, which he tells us were then abroad, see sects. 4 and 8, the Theudas of the Acts might be at the head of one of those seditions, though not particularly named by him. This he informs us here, sect. 6, and War 2.4.2, that certain of the seditious came and burnt the royal palace at Amathus, or Betharamphta, upon the river Jordan. Perhaps their leader, who is not named by Josephus, might be this Theudas."
This doesn't give us an exact answer, but I think the approach is right. Because Acts says that there was a Theudas prior to Judas, we know that the Theudas Josephus refers to is a different Theudas or else Jospehus is simply confused in his chronology.
Acts 6:9 mentions the Province of Cilicia during a scene allegedly taking place in mid-30s AD. The Roman province by that name had been on hiatus from 27 BC and re-established by Emperor Vespasian only in 72 AD.
Most Bible dictionaries should point out that there were regions referred to as Cilicia prior to their being unified into a single province by Vespasian. Specifically, there was Cilicia Tracheia and Cilicia Campestris (Pedias) which Vespesian joined together into the single province. So there is no problem in referring to the region of Cilicia. The Anchor Yale Bible Dictionary, for instance, notes that "References in the NT to Cilicia consistently link it to Syria reflecting the administrative unity of these two areas during the early principate (Acts 15:23, 41; Gal 1:21; also Acts 27:5)."
Acts 4:4 speaks of Peter addressing an audience of 5,000 people. Professor of New Testament Robert M Grant says: 'Luke evidently regarded himself as a historian, but many questions can be raised in regard to the reliability of his history […] His ‘statistics’ are impossible; Peter could not have addressed three thousand hearers without a microphone, and since the population of Jerusalem was about 25-30,000, Christians cannot have numbered five thousand
Why couldn't Christians have numbered five thousand? The simple fact that that there existed 25-30,000 people does not in itself tell us how many of them could have been Christians. Luke repeatedly attributes the growth in numbers to the work of God (2:41; 2:47; 5:14; 11:24), so I have no reason to think a high number of converts is implausible. And, as another commenter rightly noted, the population of Jerusalem greatly increased around festivals (it grew to over 100,000 around Passover). It's not necessary to think each person being saved was an actual resident of Jerusalem.
Concerning whether or not Peter could have spoken to 3,000 people without a mircrophone, that would depend on a number of different factors, mainly his environment and positioning within the environment and the speaker himself. (And I think Charis provides us with a good possibility.)
Consider this interesting passage from Benjamin Franklin's autobiography (he was friends with George Whitefield, though not himself Christian, and heard him preach on several occasions):
"He [Whitefield] had a loud and clear voice, and articulated his words and sentences so perfectly, that he might be heard and understood at a great distance, especially as his auditories, however numerous, observ'd the most exact silence. He preach'd one evening from the top of the Court-house steps, which are in the middle of Market-street, and on the west side of Second-street, which crosses it at right angles.
Both streets were fill'd with his hearers to a considerable distance. Being among the hindmost in Market-street, I had the curiosity to learn how far he could be heard, by retiring backwards down the street towards the river; and I found his voice distinct till I came near Front-street, when some noise in that street obscur'd it. Imagining then a semi-circle, of which my distance should be the radius, and that it were fill'd with auditors, to each of whom I allow'd two square feet, I computed that he might well be heard by more than thirty thousand. This reconcil'd me to the newspaper accounts of his having preach'd to twenty-five thousand people in the fields, and to the antient histories of generals haranguing whole armies, of which I had sometimes doubted."
So, again, I don't see anything implausible in Peter speaking to an audience of a several thousand (even if Franklin is exaggerating a bit).
In Acts 21:38, a Roman asks Paul if he is 'the Egyptian' who led a band of 'sicarii' (literally: 'daggers') into the desert. In both The Jewish Wars and Antiquities of the Jews, Josephus talks about Jewish nationalist rebels called sicarii directly prior to talking about The Egyptian leading some followers to the Mount of Olives. Richard Pervo believes that this demonstrates that Luke used Josephus as a source and mistakenly thought that the sicarii were followers of The Egyptian.The hypothesis that Luke-Acts used Josephus as a source is rare.
First, it's also possible that the Egyptian did gather sicarii as his followers and that Josephus didn't know about it or simply failed to specify this.
Second, Luke records the statement of the Roman. It's possible that the Roman believed sicarii were among the followers of the Egyptians and Luke is simply faithfully recording what the Roman said.
Acts 10:1 speaks of a Roman Centurion called Cornelius, stationed in Caesarea. But during the reign of Herod Agrippa, no Roman troops were stationed in his territory.
According to the IVP Bible Back. Comm. "Five cohorts were stationed in Caesarea and another in Jerusalem. This particular cohort is known to have been in Palestine in A.D. 69, though not specifically attested here in this period; from A.D. 41 to 44 Agrippa I had his own troops in Caesarea, so these events must be before A.D. 41" and the Zonder. Illust. Bible Back. Comm. (an excellent resource, by the way) remarks that this is probably circa A.D. 40 and that it is even possible that the Centurion is retired (as he seems well established in the city, has a house and his family with him).
The description of the 'Apostolic Council' in Chapter 15 has been described as 'an imaginative construction [that] corresponds to no historical reality’. Paul's trial in Acts 24 has been described as 'incoherently presented'
These are just assertions that should have no value for persons who don't already agree. After all, atheists think God is an imaginative construction corresponding to no historical reality and they find the case for God to be incoherently presented. But I don't find that assertion very convincing either.
