The decay $\bar{B} \rightarrow D^* \tau^- \bar{\nu}_\tau$ is sensitive to new physics with a non-universal coupling over the three generation of the leptonic sector. We report the first measurement of the $\tau$ lepton polarization $P_\tau(D^*)$ and a new measurement of the ratio of the branching fractions $R(D^*) = \mathit{BF}(\bar{B} \rightarrow D^* \tau^- \bar{\nu}_\tau) / \mathit{BF}(\bar{B} \rightarrow D^* \ell^- \bar{\nu}_\ell)$, where $\ell^-$ denotes $e^-$ or $\mu^-$, using the full data sample containing $(7.72 \pm 0.11) \times 10^8$ $B \bar{B}$ pairs accumulated at the Belle experiment. We reconstruct signal events from $\tau^- \rightarrow \pi^- \nu_\tau$ and $\rho^- \nu_\tau$. Our measurement results in $P_\tau(D^*) = -0.38 \pm 0.51 ({\rm stat.}) ^{+0.21}_{-0.16} ({\rm syst.})$ and $R(D^*) = 0.270 \pm 0.035 ({\rm stat.}) ^{+0.028}_{-0.025} ({\rm syst.})$. These are compatible with the SM prediction within 0.4$\sigma$.