The catalytic oxidation of methane to carbon dioxide and water over PdO(101) is investigated with first-principles based microldnetic modeling. Extensive exploration of the reaction landscape allows for determination of preferred pathways at different reaction conditions. The predicted kinetic behavior is in good agreement with a range of experimental findings including reaction orders in methane, water, and oxygen as well as apparent activation energies. The results consolidate the role of the PdO(101) surface in the activity of PdO catalysts and offer starting points for computational design of materials with improved catalytic activity. Moreover, the study demonstrates the predictive power of first-principles based kinetic modeling for oxide surfaces when hybrid functionals are applied in conjugation with kinetic models that go beyond the mean-field approximation.