It is well-known that light bridges inside a sunspot produce small-scale plasma ejections and transient brightenings in the chromosphere, but the nature and origin of such phenomena are still unclear. Utilizing the high-spatial and high temporal resolution spectral data taken with the Fast Imaging Solar Spectrograph and the TiO 7057 A broadband filter images installed at the 1.6 meter New Solar Telescope of Big Bear Solar Observatory, we report arcsecond-scale chromospheric plasma ejections (1.7 arc) inside a light bridge. Interestingly, the ejections are found to be a manifestation of upwardly propagating shock waves as evidenced by the sawtooth patterns seen in the temporal-spectral plots of the Ca II 8542 A and H-alpha intensities. We also found a fine-scale photospheric pattern (1 arc) diverging with a speed of about 2 km/s two minutes before the plasma ejections, which seems to be a manifestation of magnetic flux emergence. As a response to the plasma ejections, the corona displayed small-scale transient brightenings. Based on our findings, we suggest that the shock waves can be excited by the local disturbance caused by magnetic reconnection between the emerging flux inside the light bridge and the adjacent umbral magnetic field. The disturbance generates slow-mode waves, which soon develop into shock waves, and manifest themselves as the arcsecond-scale plasma ejections. It also appears that the dissipation of mechanical energy in the shock waves can heat the local corona. ; Comment: 27 pages, 9 figures, accepted for publication in ApJ