We present the first Suzaku observation of the new Soft Gamma Repeater SGR 0501+4516, performed on 2008 August 26, four days after the onset of bursting activity of this new member of the magnetar family. The soft X-ray persistent emission was detected with the X-ray Imaging Spectrometer (XIS) at a 0.5-10 keV flux of 3.8E-11 erg/s/cm2, with a spectrum well fitted by an absorbed blackbody plus power-law model. The source pulsation was confirmed at a period of 5.762072+/-0.000002 s, and 32 X-ray bursts were detected by the XIS, four of which were also detected at higher energies by the Hard X-ray Detector (HXD). The strongest burst, which occurred at 03:16:16.9 (UTC), was so bright that it caused instrumental saturation, but its precursor phase, lasting for about 200 ms, was detected successfully over the 0.5-200 keV range, with a fluence of ~2.1E-7 erg/cm2 and a peak intensity of about 89 Crab. The entire burst fluence is estimated to be ~50 times higher. The precursor spectrum was very hard, and well modeled by a combination of two blackbodies. We discuss the bursting activity and X/gamma-ray properties of this newly discovered Soft Gamma Repeater in comparison with other members of the class.