Single particle dynamics of water confined in a nanopore is studied through computer molecular dynamics. The pore is modeled to represent the average properties of a pore of Vycor glass. Dynamics is analyzed at different hydration levels and upon supercooling. At all hydration levels and all temperatures investigated a layering effect is observed due to the strong hydrophilicity of the substrate. The time density correlators show, already at ambient temperature, strong deviations from the Debye and the stretched exponential behavior. Both on decreasing hydration level and upon supercooling we find features that can be related to the cage effect typical of a supercooled liquid undergoing a kinetic glass transition. Nonetheless the behavior predicted by mode coupling theory can be observed only by carrying out a proper shell analysis of the density correlators. Water molecules within the first two layers from the substrate are in a glassy state already at ambient temperature (bound water). The remaining subset of molecules (free water) undergoes a kinetic glass transition; the relaxation of the density correlators agree with the main predictions of the theory. From our data we can predict the temperature of structural arrest of free water.