Curl at a given point can be defined as $\nabla x \vec f =( \int_{\delta A} \vec f \cdot \vec {dl}) / A$ where the area $A$ goes to 0.
Doing the specific case example around the origin, you get $(\int_{\delta A} \vec f \cdot \vec {dl}) / A= (\int_0^{2 \pi} d \theta \, r f_\theta)/ \pi r^2$
$$= 2 \pi r f_\theta / \pi r^2 = 2 f_\theta/r$$
This is not the same as the formula for the z component in cylindrical coordinates, $$f_\theta/r $$
Why is my answer twice the correct one?
On
You have chosen a field such that, at a particular radius (let's call it $r_{0}$), $f_{\theta}$ = constant = $Cr_{0}$ and the other components are 0.
You have assumed that the curl is constant throughout the area $A$; the general form is given by $\int { \left( \nabla \times \overrightarrow { f } \right) \cdot dA } =\oint { \overrightarrow { f } \cdot d\overrightarrow { l } } $ (see the expression for curl in cylindrical coordinates in https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates). The curl is constant only when the $\theta$ component of the field varies as $\propto { r }$. Then, plugging $\overrightarrow { f } =\left( \begin{matrix} 0 \\ 0 \\ Cr \end{matrix} \right) $ into the expression for the curl in cylindrical coordinates (which is in the aforementioned Wikipedia link) gives $\left| \nabla \times \overrightarrow { f } \right| =2C$, which gives the factor of 2 present in your answer.
I think you calculate the curl in 0. But, at this point, the field is not defined if $f_\theta=cst$
Have you tried to represent this field at the origin 0 ?