It would seem that this problem could be re-expressed in a manner that makes the form of the answer intuitively clear.
Consider a cylinder oriented vertically. Its top (disk-shaped) surface is coated with a thin layer of "North" magnetic monopoles. Its bottom surface is similarly coated with "South" magnetic monopoles. The spinning of the cylinder along its vertical axis creates rings of magnetic current due to the resulting circular paths traced out by the magnetic monopoles.
These magnetic currents appear in the Maxwell equation corresponding to Faraday's Law in a manner exactly analogous to the appearance of electric currents in the Maxwell equation corresponding to Ampere's Law. [This term in the Faraday's Law equation is normally zero, because there are no magnetic monopoles, hence no magnetic currents.]
The rings of magnetic current will produce toroidal electric fields (concentrated at the ends of the cylinder). These fields are analogous to the toroidal magnetic fields produced by rings of electric current.
The validity of this answer (that E is not zero, but a toroidal configuration at both ends) depends on whether a magnetic dipole formed from the aforementioned distribution of magnetic monopoles is equivalent to your cylindrical magnet.
Interestingly, although magnetic monopoles could be used to create a magnetic field identical to that of your magnetic (whose field result from electric charges circulating about nuclei), the situations are not equivalent. There are only two ways to generate an electric field -- electric charges or magnetic currents. Spinning the magnet does not spontaneously generate electric charge. Nor, in the absence of mononoples, does it create a magnetic current.
By the way, Gauss' Law does not imply that the electric field is zero. It only implies that the integral of the electric field on a surface enclosing the cylinder is zero. [As pointed out by Edward, for an electric dipole this integral is zero, but the electric field itself is not.]
