is the Sudoku puzzle NP-complete?

The 9x9 board cannot be NP-complete, because there are finitely many instances of the problem.

Rather than strict NP-completeness, what I think what you're trying to ask is whether there is a transition similar to the one between 3-SAT and 2-SAT where the problem goes from being NP-hard to easy and in P. There is such a transition, but 9x9 Sudoku is on the wrong side of it. Solving a Sudoku puzzle is equivalent to deciding whether there is a valid graph vertex coloring using $k$ colors, where $k = n^2$ in your $n^2 \times n^2$ Sudoku instance. The graph coloring problem is known to be NP-complete for values of $k$ > 2, so 9x9 Sudoku is still hard.