Since line $l_1: x + 2y - 1 = 0$ and line $l_2: mx - y = 0$ are perpendicular to each other,
we can obtain $m - 2 = 0$. Solving this equation yields $m = 2$.
Therefore, the correct choice is: $\boxed{\text{A}}$.
By establishing an equation about $m$ based on the condition that the two lines are perpendicular to each other, solving it will yield the value of the real number $m$.
This problem provides an equation of a line containing the letter parameter $m$, and asks for the value of $m$ under the condition that they are perpendicular to each other. It focuses on the necessary and sufficient condition for two lines to be perpendicular, making it a basic question.