“Now in the first year of Cyrus king of Persia.” Ezra 1:1-3a is almost identical to the last two verses of 2 Chronicles. From a chronological point of view, Ezra takes over where Chronicles stops.
This first year of Cyrus refers to his first year reigning over Babylon, which he conquered in 538 BC, not his first year reigning in Persia (559 BC). Babylon was captured in October, 539 BC (Dan. 5:30-31), but Cyrus did not get there and start to reign over Babylon until March of 538 BC. Quite a lot is known about Cyrus from the Persian records. Also, “Cyrus” is an actual name, not a title, whereas in other parts of Ezra, Nehemiah, and Esther, the “names” Darius, Xerxes, Artaxerxes, and Ahasuerus are actually titles, not proper names.
“by the mouth of Jeremiah.” Jeremiah had prophesied that the Babylonian Captivity would be 70 years (Jer. 25:11-12; 29:10), so Yahweh worked behind the scenes to make sure that prophecy came to pass.