Stardate 2233.04=January 4, 2233
Probably not. After all, 2233.04 must be the same as 2233.040 if we're going to have three digits past the decimal point later in the year anyway.
And it's not as if Kirk could be unsure about the proper day by dozens of days! Yet he is in doubt about the second digit past the comma when doing his Delta Vega castaway log. So it apparently stands for days at most, and more probably for hours.
So the only thing about the system that really makes "more" sense than the older stardate systems is that the first four digits spell the year - which was the dramatic purpose of the entire change, so that Nero could query Robau about the year.
And that still makes very little sense, because Nero would glean very little information content from what Robau says, quite regardless of whether he uses plain Earth years or ancient Earth stardates. It's only for the audience, but doesn't work in the in-universe context at all. Something on the lines of "Why, it's stardate 6523.2, of course" followed by Nero and his lieutenant discussing this revelation in terms of actual years (perhaps in subtitled Romulan) would have made logical sense, but would not have been dramatically as simplistic as the rest of the movie, and would have ill served the scene.
Timo Saloniemi