Josephus refers to more than one man named Jesus who died by Roman hands in these days. The OP refers to one of them as "Jesus # 2." If I guess correctly at the OP's meaning he was one of the leading priests in Jerusalem during the Revolt of 66-70, who lost his life to Vespasian's forces under Titus. He was not Jesus of Nazareth and the exact manner of his death is not specified. This Jesus is introduced in chapter 4 of the Wars of the Jews and his death is described in chapter 5:
To say all in a word, if Ananus [the high priest] had survived, they would have certainly
compounded matters... And the Jews would have then put an abundance
of delays in the way of the Romans, if they had had such a general as
he was. Jesus [the elder priest] was also joined with him... I cannot
but think that it was because God had doomed this city to destruction,
as a polluted city, and was resolved to purge his sanctuary by fire,
that he cut off these their great defenders and well-wishers, while
those that a little before had worn the sacred garments, and had
presided over the public worship; and had been esteemed venerable by
those that dwelt on the whole habitable earth when they came into our
city, were cast out naked, and seen to be the food of dogs and wild
beasts... And this at last was the end of Ananus and Jesus.
It is understandable that Christians confuse one Jesus with another, but this Greek rendering of a Hebrew name was actually quite common. No crucifixion is mentioned here, however.