Regarding the similarities between Saulos and Paul, both (at least arguably, in Paul's case) were related to the Herodians.
War 2.17.4:
So the men of power perceiving that the sedition was too hard for them to subdue, and that the danger which would arise from the Romans would come upon them first of all, endeavored to save themselves, and sent ambassadors ... among whom the most eminent were Saul, and Antipas, and Costobarus, who were of the king's kindred; and they desired of them both that they would come with an army to the city, and cut off the seditious before it should be too hard to be subdued.
Rom. 16:10-11:
Greet those who belong to the household of Aristobulus. Greet my kinsman Herodion. Greet those in the Lord who belong to the family of Narcissus.
Watson notes that this Aristobulus is "often identified" with the Herodians and that the one who was married to Salome had a son named Herod during Paul's time.
https://books.google.com/books?id=4PfFg ... me&f=false
And den Hollander notes that it is argued that this son "is also the 'Herodion' within the household of Aristobulus to whom Paul directs greetings in Rom. 16:10-11."
https://books.google.com/books?id=TPjYA ... me&f=false
Bird talks about Aristobulus and Herodion and their possible connection to the Herodian dynasty here (though he sees Herodion as a Herodian slave rather than as Aristobulus's son).
https://books.google.com/books?id=pU0VC ... ul&f=false
Regarding the identity of Herodion and whether he was Paul's relative or countryman, Holland writes, "The greeting gives little away about Herodion. Indeed, the virtual silence supports the view that he was a blood relative as there seems to be little point in Paul isolating one of his many countrymen in Rome in order to send such a brief greeting."
https://books.google.com/books?id=ZblMA ... er&f=false
And both Saulos and Paul were associated with Nero and Achaia and were anti-sedition.
War 2.17.4:
So the men of power perceiving that the sedition was too hard for them to subdue, and that the danger which would arise from the Romans would come upon them first of all, endeavored to save themselves, and sent ambassadors, some to Florus, the chief of which was Simon the son of Ananias; and others to Agrippa, among whom the most eminent were Saul, and Antipas, and Costobarus, who were of the king's kindred; and they desired of them both that they [Florus and Agrippa] would come with an army to the city, and cut off the seditious before it should be too hard to be subdued.
War 2.20.1:
Costobarus, therefore, and Saul, who were brethren, together with Philip, the son of Jacimus, who was the commander of king Agrippa's forces, ran away from the city, and went to Cestius. But then how Antipas, who had been besieged with them in the king's palace, but would not fly away with them, was afterward slain by the seditious, we shall relate hereafter. However, Cestius sent Saul and his friends, at their own desire, to Achaia, to Nero, to inform him of the great distress they were in, and to lay the blame of their kindling the war upon Florus, as hoping to alleviate his own danger, by provoking his indignation against Florus.
Php. 4:22:
All the saints greet you, especially those of Caesar’s household.
Rom. 13:1-5:
Let everyone be subject to the governing authorities, for there is no authority except that which God has established. The authorities that exist have been established by God. Consequently, whoever rebels against the authority is rebelling against what God has instituted, and those who do so will bring judgment on themselves ... Therefore, it is necessary to submit to the authorities, not only because of possible punishment but also as a matter of conscience.
Rom. 15:26:
For Macedonia and Achaia were pleased to make a contribution for the poor among the Lord's people in Jerusalem.
1 Cor. 16:15:
You know that the household of Stephanas were the first converts in Achaia, and they have devoted themselves to the service of the Lord's people.
2 Cor. 1:1:
Paul, an apostle of Christ Jesus by the will of God, and Timothy our brother, to the church of God in Corinth, together with all his holy people throughout Achaia.
1 Thes. 1:7-8:
... you have become an example to all the believers in Macedonia and Achaia. For not only did the message of the Lord ring out from you to Macedonia and Achaia, but your faith in God has gone out to every place, so that we have no need to say anything further.
And both Saulos and Paul were violent.
Ant. 20.9.4 (after the James passage in 20.9.1):
Costobarus also, and Saulus, did themselves get together a multitude of wicked wretches, and this because they were of the royal family; and so they obtained favor among them, because of their kindred to Agrippa; but still they used violence with the people, and were very ready to plunder those that were weaker than themselves. And from that time it principally came to pass that our city was greatly disordered, and that all things grew worse and worse among us.
Gal. 1:13:
For you have heard of my previous way of life in Judaism, how intensely I persecuted the church of God and tried to destroy it.
You know in spite of all you gained, you still have to stand out in the pouring rain.