The usual, humorous, phrase is more like this:
My daughter is 16, going on 32!
It means she is precocious. She is only 16, but she already behaves in a very mature way.
Though the usual, humorous, phrase is "16 going on 32" (or another combination going upwards by a huge step), the author here is deliberately turning it around the other way.
So the author here is expressing, humorously, that the woman is 22, but behaves very immaturely.
It's a very common - indeed, overused - "trick" in English commercial writing in the present era, to "turn around" a common humorous phrase - making it "even more" humorous, if you will. This is an example of that.
Note that as Robusto explains, "going on" very simply means "almost". For example, "to walk to the store is five, going on six, miles", "renovation costs are 80 thousand, going on 90 thousand."
So, to recap. To really "get" the feel of this in English, relies on this chain:
1) Straightforward use of "going on": the child is six going on seven. (Simply means nothing more than "almost seven".
2) Very commonplace "humorous" use of "going on" with a large gap, used specifically of precocious children: that girl is 15 going on 35!
3) In this case, the author has "turned around" that usual humorous pattern: "the person is 35 going on 15". Note that it is common (today) in English to invert a common humorous construction, to create a (supposedly) even funnier one.