The Verdict: Need More Info
You can apply once you turn 18; the New Zealand arrangement is open to ages 18 to 30. Come back then and book through the Embassy of Mexico in Wellington.
The age is measured when you apply, so you will have a wide window once it opens.
If you plan to study in Mexico in the meantime, student residence has no age limit and allows some work at university level.
And if a parent or grandparent was born in Mexico, you may be Mexican already and need no visa at all.