We report on our second-year campaign of X-ray follow-up observations of unidentified Fermi-LAT \gamma-ray sources at high Galactic latitudes (|b|>10 degree) using the X-ray Imaging Spectrometer onboard the Suzaku X-ray Observatory. In this second year of the project, seven new targets were selected from the First Fermi-LAT Catalog, and studied with 20-40 ks effective Suzaku exposures. We detected an X-ray point source coincident with the position of the recently discovered millisecond pulsar PSR J2302+4442 within the 95% confidence error circle of 1FGL J2302.8+4443. The X-ray spectrum of the detected counterpart was well fit by a blackbody model with temperature of kT ~0.3 keV, consistent with an origin of the observed X-ray photons from the surface of a rotating magnetized neutron star. For four other targets which were also recently identified with a normal pulsar (1FGL J0106.7+4853) and millisecond pulsars (1FGL J1312.6+0048, J1902.0-5110, and J2043.2+1709), only upper limits in the 0.5-10 keV band were obtained at the flux levels of ~10^{-14} erg cm^{-2} s^{-1}. A weak X-ray source was found in the field of 1FGL J1739.4+8717, but its association with the variable \gamma-ray emitter could not be confirmed with the available Suzaku data alone. For the remaining Fermi-LAT object 1FGL J1743.8-7620 no X-ray source was detected within the LAT 95% error ellipse. We briefly discuss the general properties of the observed high Galactic-latitude Fermi-LAT objects by comparing their multiwavelength properties with those of known blazars and millisecond pulsars.