TL;DR: It does seem contradictory. However, your interpretation doesn't fit the context of that first song. It's much more likely that "Two and twenty" refers to the age-year he's in: having turned 21, he's now (by definition) in his 22nd year.
If I understand you correctly, you suggest that "Two and twenty, now he’s rising" is a multiply-reversed way of saying "now he's rising 22". You suggest one possible interpretation, in which "rising" is equivalent to "on course for" (i.e. heading towards or approaching, in the same way a rising tide approaches the peak). Thus, Frederic is 21 and "rising" 22.
However, that interpretation seems unwieldy in the context of the lyrics. Here's the opening of the song:
Pour, oh, pour the pirate sherry
Fill, O fill the pirate glass!
And, to make us more than merry
Let the pirate bumper pass!
For today our pirate 'prentice
Rises from indenture freed
Strong his arm, and keen his scent is
He's a pirate now indeed!
Here's good luck to Frederic's ventures!
Frederic's out of his indentures
Two and twenty, now he's rising
And alone he's fit to fly
Which we're bent on signalizing
With unusual revelry.
Frederic is rising from his indenture as a pirate and (continuing the metaphor) is now fit to fly alone. "Two and twenty, now he's rising" means "He's 22 and now he's rising (to the level of full pirate)". It's a mistake to see "two and twenty" as the complement of "now he's rising"; instead, "now he's rising" is intended to be read with the next line – "now he's rising and alone he's fit to fly".
In Act II, you reference the song "When You Had Left Our Pirate Fold", which is a key turning point in the play. Without giving away the plot, it's sufficient to say that the interpretation of Frederic's age is a critical matter. Here's the relevant text:
I'm afraid you don't appreciate the delicacy of your position
You were apprenticed to us!
[Frederic] Yes, until I reached my twenty-first year
No, no, no
Until you reached your twenty-first birthday!
Since everyone is unequivocal that Frederic had reached his twenty-first year (and therefore, they had all assumed, had completed his apprenticeship as a pirate), the only possible explanation for the earlier "two and twenty" reference is that having turned 21, Frederic is now in his 22nd year.
It's beyond the realms of possibility that "two and twenty" is a mistake. Perhaps Sir William Schwenck Gilbert simply found the alliteration more appealing than "one and twenty"; or perhaps he was just being perverse.