Inclusive J/ψ production is studied in p-Pb interactions at a centre-of-mass energy per nucleon-nucleon collision sNN=8.16 TeV, using the ALICE detector at the CERN LHC. The J/ψ meson is reconstructed, via its decay to a muon pair, in the centre-of-mass rapidity intervals 2.03 < y_cms < 3.53 and −4.46 < y_cms < −2.96, where positive and negative y_cms refer to the p-going and Pb-going direction, respectively. The transverse momentum coverage is p_T < 20 GeV/c. In this paper, y_cms- and p_T-differential cross sections for inclusive J/ψ production are presented, and the corresponding nuclear modification factors R_pPb are shown. Forward results show a suppression of the J/ψ yield with respect to pp collisions, concentrated in the region p_T ≲ 5 GeV/c. At backward rapidity no significant suppression is observed. The results are compared to previous measurements by ALICE in p-Pb collisions at sNN=5.02 TeV and to theoretical calculations. Finally, the ratios R_FB between forward- and backward-y_cmsR_pPb values are shown and discussed.[Figure not available: see fulltext.].