The problem has long been noted by commentators. Here, for example, is Rashi's discussion:
the twentieth year: This refers to the twentieth year of King Darius, who is identical with Artaxerxes. In Tractate Rosh Hashanah (3a, b), we find that the phrase “the twentieth year” is mentioned twice using similar wording [to teach us that it is speaking of the identical year]. It states below (2:1), “And it came to pass in the month of Nissan of the twentieth year of Artaxerxes, etc.” The years were calculated from the month of Tishrei. It says here, “And it was in the month of Kislev of the twentieth year, etc.” This refers to the Kislev coming after Tishrei. Further down it says (2:1), “And it was in the month of Nissan of the twentieth year, etc.” This refers to the Nissan of the same year. And I cannot explain that they calculated their years from the month of Nissan and that the incident mentioned below: “And it was in the month of Nissan, etc.” occurred before this incident: “And it was in the month of Kislev, etc.” It is true that events [in the Torah] are not necessarily recorded in chronological order. Nevertheless, the chapters indicate that the second chapter took place only because of the events of the first chapter.
His admission of puzzlement at the discrepancy between Neh 1:1 and 2:1 is plainly stated: "And I cannot explain that they calculated their years from the month of Nissan and that the incident mentioned below ... occurred before this incident..."!
I think just about every commentator takes a stab at it (I confess I haven't checked them all!). So far as I'm aware, the fullest discussion is by H.G.M. Williamson, Ezra, Nehemiah (Word, 1985), pp. 168-170. It isn't possible to reproduce three pages of discussion here. But Williamson offers these as the main explanations, "though not all are equally plausible" (p. 169):
- Nehemiah was using a calendar with the new year marked from Tishri. This is the suggestion of Edwin Thiele (Mysterious Numbers of the Hebrew Kings [Paternoster, 1966], p. 30). But not only is this confusing, but later studies have shown Thiele's suggestion to be unlikely in any case.
- As a variation, some theorize that we have already in evidence the later Jewish custom of an autumnal new year, but again it is highly unlikely in Nehemiah's day.
- Of course, textual emendation is always an option! And it was exercised by Wilhelm Rudolph in his major Nehemiah commentary (1949), p. 102, who suggested that the Hebrew read "in the nineteenth year of Artaxerexes the king", but his explanation for how this textual situation came about does not convince many.
- There are other, yet more far-fetched rewritings of 1:1, but I omit these.
Some suggest that these date formulae relate to regnal rather than calendar years. This has been defended by Elias Bickerman, "En marge de l'Écriture. I. - Le comput des années de règne des Achéménides...," Revue Biblique 88 (1981): 19-23.
1 His argument runs as follows (quoting Williamson's summary) -
[Bickerman] argues that in court circles, in which Nehemiah moved, it would not have been unusual to follow the regnal rather than the calendar year, and he cites an independent example from Thucydides (8:58) to support this claim... [other data omitted] ... Thus 1:1 would refer to December 446 BC and 2:1 to the following spring.
Williamson finds Bickerman's solution "the neatest explanation", provided that his "arguments are sound" -- and they appear to be. So, following Bickerman, Williamson concludes (p. 170):
In so personal a composition, Nehemiah may be allowed to have used the calendar most familiar to himself, since he was not primarily addressing his Jewish compatriots in his memoir. The best alternative solution is one which sees the confusion arising at some [editorial] stage in the handling of the material.
1 Those interested in Bickerman's work might also wish to consult: W. Kendrick Pritchett, "Thucydides' Statement on His Chronology," Zeitschrift für Papyrologie und Epigraphik 62 (1986): 205-11. The first page (free to view) summarizes Bickerman nicely.