The effect of pressure on the crystalline structure and superconducting transition temperature (T-c) of the 111-type Na1-xFeAs system using in situ high-pressure synchrotron X-ray powder diffraction and diamond anvil cell techniques is studied. A pressure-induced tetragonal to tetragonal isostructural phase transition was found. The systematic evolution of the FeAs4 tetrahedron as a function of pressure based on Rietveld refinements on the powder X-ray diffraction patterns was obtained. The nonmonotonic T-c(P) behavior of Na1-xFeAs is found to correlate with the anomalies of the distance between the anion (As) and the iron layer as well as the bond angle of As-Fe-As for the two tetragonal phases. This behavior provides the key structural information in understanding the origin of the pressure dependence of T-c for 111-type iron pnictide superconductors. A pressure-induced structural phase transition is also observed at 20 GPa.