We present an analysis of the globular cluster (GC) system of the nucleated dwarf elliptical galaxy VCC 1087 in the Virgo Cluster based on Keck LRIS spectroscopy and archival Hubble Space Telescope Advanced Camera for Surveys imaging. We estimate that VCC 1087 hosts a total population of 77+/-19 GCs, which corresponds to a relatively high V-band specific frequency of 5.8+/-1.4. The g475-z850 color distribution of the GCs shows a blue (metal-poor) peak with a tail of redder (metal-rich) clusters similar in color to those seen in luminous elliptical galaxies. The luminosity function of the GCs is lognormal and peaks at MTOg475=-7.2+/-0.3, MTOz850=-8.1+/-0.2. These peak positions are consistent with those found for luminous Virgo elliptical galaxies, suggesting either the lack of or, surprisingly similarly, the dynamical destruction processes of GCs among dwarf and giant galaxies. Spectroscopy of a subsample of 12 GCs suggests that the GC system is old and coeval (>~10 Gyr), with a fairly broad metallicity distribution (-1.8<~[M/H]<~-0.8). In contrast, an integrated spectrum of the underlying galaxy starlight reveals that its optical luminosity is dominated by metal-rich, intermediate-age stars. The radial velocities of the GCs suggest rotation close to the major axis of the galaxy, and this rotation is dynamically significant with (vrot/σlos)*>1. A compilation of the kinematics of the GC systems of nine early-type galaxies shows surprising diversity in the (vrot/σlos) parameter for GC systems. In this context, the GC system of VCC 1087 exhibits the most significant rotation-to-velocity dispersion signature. Dynamical mass modeling of the velocity dispersion profile of the GCs and galaxy stars suggests fairly constant mass-to-light ratios of ~3 out to 6.5 kpc. The present observations can entertain both baryonic and nonbaryonic solutions, and GC velocities at larger radii would be most valuable with regard to this issue. Finally, we discuss the evolution of VCC 1087 in terms of the galaxy ``harassment'' scenario and conclude that this galaxy may well be the remains of a faded, tidally perturbed Sc spiral. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.