Among all hydrocarbons, the methane molecule contains the highest amount of hydrogen with respect to carbon. Therefore, the catalytic decomposition of methane is considered as an efficient approach to produce hydrogen along with nanostructured carbon product. On the other hand, the presence of hydrogen in the composition of the initial gas mixture is required for the stable operation of the catalyst. In present work, the experiments on the catalytic decomposition of methane–hydrogen mixture were performed in a flow-through quartz reactor equipped with McBain balances under atmospheric pressure. The catalyst NiO-CuO/Al2O3 was prepared by the mechanochemical activation technique. The maximum carbon yield of 34.9 g/gcat was obtained after 2 h of experiment at 610 °C. An excess of hydrogen in the reaction mixture provided the long-term activity of the nickel–copper catalyst. The durability tests ongoing for 6 h within a temperature range of 525–600 °C showed no noticeable deactivation of the catalyst. Two kinetic models, D1a and M1a, were proposed for the studied decomposition of the methane–hydrogen mixture over the nickel–copper catalyst. The kinetic constants for these models were determined by means of mathematical modelling.