Band parameters and superconductivity of yttrium hypocarbide (Y2C) have been investigated. The computations are performed using first-principles pseudopotential method within a generalized gradient approximation. The equilibrium lattice parameters have been determined and compared with experiment. Moreover, the material of interest is found to be stiffer for strains along the a-axis than those along the c-axis. A band-structure analysis of Y2C implied that the latter has a metallic character. The examination of Eliashberg Spectral Function indicates that Y-related phonon modes as well as C-related phonon modes are considerably involved in the progress of scattering of electrons. By integrating this function, the value of the average electron-phonon coupling parameter (lambda) is found to be 0.362 suggesting thus that Y2C is a weak coupling Bardeen-Copper-Schrieffer superconductor. The use of a reasonable value for the effective Coulomb repulsion parameter (mu* = 0.10) yielded a superconducting critical temperature T-c of 0.59 K which is comparable with a previous theoretical value of 0.33 K. Upon compression (at pressure of 10 GPa) lambda and T-c are increased to be 0.366 and 0.89 K, respectively, showing thus the pressure effect on the superconductivity in Y2C. The spin-polarization calculations showed that the difference in the total energy between the magnetic and non-magnetic Y2C is weak.