Vidqun:
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.
Josephus repeating a Jewish tradition that existed by his time is not evidence that Cyrus or Darius actually said something or why. For a start, Cyrus did not make any special edict just for the Jews, but had a broader policy of allowing captives in Babylonia to return to their homelands and practice their own religious beliefs. This was in order to quell social unrest, for the same reason that Cyrus dedicated a temple to Marduk in Babylon.
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.''
Antiquities here interpolates part of Isaiah 44:28-45:1 (not written by Isaiah) into Ezra 1:2 rather than actually quoting Cyrus. The fact that Josephus interpolated the claim about Cyrus knowing of the supposed 'prophecy' (or quoted a Jewish tradition that had developed after the writing of Ezra) into the statement in Ezra is evident from the fact that the 'quote' in Antiquities isn't even consistent with what Ezra 1:1 says Cyrus proclaimed, and neither is consistent with Cyrus' broader policy rather than special treatment of just the Jews.