Overall Josephus was a trustworthy historian. According to him Cyrus was aware of the Isaiah prophecy. This made him favor the Jews, allowing them to return to their homeland and rebuild the temple. Later Darius discovered Cyrus' edict and he allowed the Jews to carry on with their work. In the end the temple was rebuilt.
JOE Antiquities of the Jews 11:1 In the first year of the reign of Cyrus {a} which was the seventieth from the day that our people were moved out of their own land into Babylon, God pitied the captivity and calamity of these poor people, according as he had foretold to them by Jeremiah the prophet, before the destruction of the city,
2 that after they had served Nebuchadnezzar and his posterity, and after they had undergone that servitude seventy years, he would restore them again to the land of their fathers, and they should build their temple, and enjoy their ancient prosperity. And these things God did afford them;
3 for he stirred up the mind of Cyrus, and made him write this throughout all Asia:--``Thus says Cyrus the king: Since God Almighty has appointed me to be king of the habitable earth, I believe that he is that God which the nation of the Israelites worship;
4 for indeed he foretold my name by the prophets, and that I should build him a house at Jerusalem, in the country of Judea.''
5 This was known to Cyrus by his reading the book which Isaiah left behind him of his prophecies; for this prophet said that God had spoken thus to him in a secret vision:--``My will is, that Cyrus, whom I have appointed to be king over many and great nations, send back my people to their own land, and build my temple.''
6 This was foretold by Isaiah one hundred and forty years before the temple was demolished. Accordingly, when Cyrus read this, and admired the divine power, an earnest desire and ambition seized upon him to fulfil what was so written; so he called for the most eminent Jews that were in Babylon, and said to them, that he gave them permission to go back to their own country, and to rebuild their city of Jerusalem, {b} and the temple of God, (Ant 11:1-6)
``King Darius to Tatnai the governor, and to Shetharboznai, sends greeting. Having found a copy of this letter among the records of Cyrus, I have sent it you; and I will that all things be done as is therein written. Farewell.''
105 So when Tatnai, and those who were with him, understood the intention of the king, they resolved to follow his directions entirely for the time to come. So they forwarded the sacred works, and assisted the elders of the Jews, and the princes of the Sanhedrin (Ant 11:104, 105 JOE)