Is this matrix surjective? Textbook dispute

Yes, this is the matrix of a surjective linear map. Look at the first and the fourth columns: $\left[\begin{smallmatrix}1\\0\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]$ respectively. It follows from this that the vectors $\left[\begin{smallmatrix}1\\0\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]$ belong to the range and that therefore the range is $k^2$, where $k$ is the field that you are working with.

Just in case, check the conventions your textbook is using, and specifically whether matrices conventionally act from left (on column vectors) or from right (on row vectors).

In particular, for your matrix $$A = \begin{bmatrix}1 & -2 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix},$$ the map $v \mapsto Av$ (where $v$ is a column vector) is surjective but not injective, while the map $u \mapsto uA$ (where $u$ is a row vector) is injective but not surjective.