  This study is aimed to contribute to a more comprehensive understanding of
the molecular hydrogen distribution in the galaxy M33 by introducing novel
methods for generating high angular resolution (18.2$''$, equivalent to 75 pc)
column density maps of molecular hydrogen ($N_{\rm H_2}$). M33 is a local group
galaxy that has been observed with Herschel in the far-infrared wavelength
range from 70 to 500 $\mu$m. Previous studies have presented total hydrogen
column density maps ($N_{\rm H}$), using these FIR data (partly combined with
mid-IR maps), employing various methods. We first performed a spectral energy
distribution fit to the 160, 250, 350, and 500 $\mu$m continuum data obtain
$N_{\rm H}$, using a technique similar to one previously reported in the
literature. We also use a second method which involves translating only the 250
$\mu$m map into a $N_{\rm H}$ map at the same angular resolution. An $N_{\rm
H_2}$ map via each method is then obtained by subtracting the HI component.
Distinguishing our study from previous ones, we adopt a more versatile approach
by considering a variable emissivity index, $\beta$ and dust absorption
coefficient, $\kappa_0$. This choice enables us to construct a $\kappa_0$ map,
thereby enhancing the depth and accuracy of our investigation of the hydrogen
column density. We address the inherent biases and challenges within both
methods (which give similar results) and compare them with existing maps
available in the literature. Moreover, we calculate a map of the carbon
monoxide CO-to-H$_2$ conversion factor ($X_\mathrm{CO}$ factor), which shows a
strong dispersion around an average value of
$1.8\times10^{20}\,\mathrm{cm^{-2}/(K\,km\,s^{-1})}$ throughout the disk. We
obtain column density probability distribution functions (N-PDFs) from the
$N_{\rm H}$, $N_{\rm H_2}$, and $N_{HI}$ maps and discuss their shape.
