This paper presents a combined numerical and analytical study of the impulsive axisymmetric spin-up from rest of an isothermal liquid metal in a closed cylinder. The motion of the liquid is caused by the action of a low-frequency, low-induction rotating magnetic field, whose magnetic Taylor number is in the range (0.01-0.9) Ta_cr^3D with Ta_cr^3D given by Grants and Gerbeth ["Linear three-dimensional instability of a magnetically driven rotating flow," J. Fluid Mech. 463, 229 (2002)]. The computations were performed for cylindrical containers of aspect ratios (diameter/height) R equal to 0.5, 1, and 2. The numerical simulations are compared with the predictions of an analytical model, valid for small Ekman numbers E extending a former work by Ungarish ["The spin-up of liquid metal driven by a rotating magnetic field," J. Fluid Mech. 347, 105 (1997)]. The first phase of the motion from rest is an initial adjustment: the inviscid fluid begins to rotate due to the externally forced azimuthal acceleration, and concomitantly a meridional velocity field is induced by the unbalanced centrifugal effects; this occurs during, approximately, the first revolution about the axis. Subsequently, the spin-up flow is dominated by an axially independent swirl in the core (which can be regarded as a geostrophic mode) and Bödewadt (Ekman) layers on the top-bottom boundaries. This is accompanied by inertial oscillations. Both the development of the swirl motion and the decay of the inertial mode occur on the spin-up time scale predicted by the asymptotic model. The investigation also points out the important role of the aspect-ratio parameter in the analysis of the magnetohydrodynamics-driven spin-up. The efficiency of the driving decays strongly for R>0.6 and becomes close to zero at R≈3.3. The numerical results confirm the analytical inference that the thickness of the sidewall layer for the angular velocity adjustment is E^1/4 /√R, and when this parameter is larger than 0.3 the sidewall influence gains dominance over the Ekman-layer effects. The frequency of the inertial oscillations increases with R.