For light waves propagating in a slab structure with a negative-zero-positive index metamaterial in the core, properties of the lateral shifts of the transmission light were investigated in detail by employing Artmann’s stationary phase method. The results show that, near the Dirac point(DP), the lateral shift can vary from positive to zero then to negative and is dozens of times even hundreds of times higher than the wavelength, which is strongly dependent on the incident angle, frequency, and the thickness of the metamaterial. For the slow wave mode, the lateral shifts and the photon tunneling were further studied. Because of the Hartman effect, the lateral shift tends to a saturation value when the barrier thickness increases, and displays different properties when the incident angle chosen near and far away from the critical angle. These results may lead to potential applications in integral optics and optical-based devices and also suggest analogous phenomena of valance electron in single-layered carbon graphene.