  We consider a non-exotic gauged U(1)$_X$ extension of the Standard Model
(SM), where the U(1)$_X$ charge of a SM field is given by a linear combination
of its hypercharge and Baryon-minus-Lepton ($B-L$) number. All the gauge and
mixed gauge-gravitational anomalies are cancelled in this model with the
introduction of three right-handed neutrinos (RHNs). Unlike the conventional
minimal U(1)$_X$ model, where a universal U(1)$_X$ charge of $-1$ is assigned
to three RHNs, we consider an alternative charge assignment, namely, two RHNs
($N_R^{1,2}$) have U(1)$_X$ charge $-4$ while one RHN ($N_R$) has a $+5$
charge. With a minimal extension of the Higgs sector, the three RHNs acquire
their Majorana masses associated with U(1)$_X$ symmetry breaking. While
$N_R^{1,2}$ have Yukawa coupling with the SM lepton doublets and play an
essential role for the 'minimal seesaw' mechanism, $N_R$ is isolated from the
SM particles due to its U(1)$_X$ charge and hence it is a natural candidate for
the dark matter (DM) without invoking additional symmetries. In this model
context, we investigate the $Z^\prime$-portal RHN DM scenario, where the RHN DM
communicates with the SM particles through the U(1)$_X$ gauge boson ($Z^\prime$
boson). We identify a narrow parameter space by combining the constraints from
the observed DM relic abundance, the results of the search for a $Z^\prime$
boson resonance at the Large Hadron Collider Run-2, and the gauge coupling
perturbativity up to the Planck/Grand Unification scale. For a special choice
of U(1)$_X$ charges for the SM fields allows us to extend the model to
SU(5)$\times$U(1)$_X$ grand unification. In this scenario, the model parameter
space is more severely constrained, which will be explored at future high
energy collider experiments.
