The invention relates to an inerter and damping controllable magneto-rheological inerter device and a control method thereof. The device comprises a cylinder, a piston, a piston rod, long and thin pipes, a thick pipe, a magneto-rheological fluid, switch valves, MR (magnetic resonance) conductors and a damping hole. According to the inerter and damping controllable magneto-rheological inerter device and the control method thereof, an apparent viscosity changing process of the magneto-rheological fluid is a continuous, stepless and reversible magneto-rheological effect under the action of an applied magnetic field, simultaneously, the low viscosity of the magneto-rheological fluid without the applied magnetic field and an inertial effect of the magneto-rheological fluid flowing in the long and thin pipes are utilized, the two effects are integrated in one device, further, the switch valves are adopted to control switching on and off of the long and thin pipes which have different pipe diameters and are filled with the magneto-rheological fluid, then the inerter is adjusted in a stepping manner,; and damping is adjusted steplessly by changing the magneto-rheological viscosity through the applied magnetic field simultaneously According to the invention, requirements for semi-active control when dynamical combination adjustment of inerter and damping is applied to a vibration isolation device for engineering are met, so that mounting and using are facilitated, the cost is reduced, and the production efficiency is improved.