Inclusive and differential fiducial cross sections of the Higgs boson are measured in the H -> ZZ* -> 4l (l = e, mu) decay channel. The results are based on proton-proton collision data produced at the Large Hadron Collider at a centre-of-mass energy of 13 TeV and recorded by the ATLAS detector from 2015 to 2018, equivalent to an integrated luminosity of 139 fb(-1). The inclusive fiducial cross section for the H -> ZZ* -> 4l process is measured to be sigma(fid) = 3.28 +/- 0.32 fb, in agreement with the Standard Model prediction of sigma(fid,SM) = 3.41 +/- 0.18 fb. Differential fiducial cross sections are measured for a variety of observables which are sensitive to the production and decay of the Higgs boson. All measurements are in agreement with the Standard Model predictions. The results are used to constrain anomalous Higgs boson interactions with Standard Model particles.