Inclusive J/$\psi$ production is studied in p-Pb interactions at a centre-of-mass energy per nucleon-nucleon collision $\sqrt{s_{\rm NN}}=8.16$ TeV, using the ALICE detector at the CERN LHC. The J/$\psi$ meson is reconstructed, via its decay to a muon pair, in the centre-of-mass rapidity intervals $2.03<y_{\rm {cms}}<3.53$ and $-4.46<y_{\rm {cms}}<-2.96$, where positive and negative $y_{\rm {cms}}$ refer to the p-going and Pb-going direction, respectively. The transverse momentum coverage is $p_{\rm T}<20$ GeV/$c$. In this paper, $y_{\rm cms}$- and $p_{\rm T}$-differential cross sections for inclusive J/$\psi$ production are presented, and the corresponding nuclear modification factors $R_{\rm pPb}$ are shown. Forward results show a suppres...