  We investigate spin and thermal transport near the N\'{e}el transition
temperature $T_N$ in three dimensions, by numerically analyzing the classical
antiferromagnetic $XXZ$ model on the cubic lattice, where in the model, the
anisotropy of the exchange interaction $\Delta=J_z/J_x$ plays a role to control
the universality class of the transition. It is found by means of the hybrid
Monte-Carlo and spin-dynamics simulations that in the $XY$ and Heisenberg cases
of $\Delta \leq 1$, the longitudinal spin conductivity $\sigma^s_{\mu\mu}$
exhibits a divergent enhancement on cooling toward $T_N$, while not in the
Ising case of $\Delta>1$. In all the three cases, the temperature dependence of
the thermal conductivity $\kappa_{\mu\mu}$ is featureless at $T_N$, being
consistent with experimental results. The divergent enhancement of
$\sigma^s_{\mu\mu}$ toward $T_N$ is attributed to the spin-current relaxation
time which gets longer toward $T_N$, showing a power-law divergence
characteristic of critical phenomena. It is also found that in contrast to the
$XY$ case where the divergence in $\sigma^s_{\mu\mu}$ is rapidly suppressed
below $T_N$, $\sigma^s_{\mu\mu}$ likely remains divergent even below $T_N$ in
the Heisenberg case, which might experimentally be observed in the ideally
isotropic antiferromagnet RbMnF$_3$.
