Thermal effects of adsorption and desorption, leading to temperature fluctuations and losses of adsorption storage systems capacity in the processes of gas charging and discharging, are the main obstacle to the wide practical application of adsorbed natural gas (ANG) technology. This work presents a numerical simulation of heat and mass transfer processes under various cyclic operation modes of a full-scale adsorption storage tank with various thermal control systems. The high-density monolithic adsorbent KS-HAM, obtained on the basis of industrial activated carbon KS-HA, was used as the adsorption material. The phase composition, surface morphology, and porous structure of the sorbents were studied. The adsorption of methane on the KS-HA adsorbent was measured. It is shown that increasing the duration of charging leads to obtaining additional capacity of the ANG system; however, the final efficiency and benefit at the end of the charging–discharging cycle are determined by the efficiency of the thermal control system and the gas-discharging mode. It has been shown that the presence of a finned thermal control system allows for charging the adsorption storage tank 3–8 times faster and provides an 8–24% greater amount of gas discharged at the discharging stage compared to the ANG system without fins.