Molecular dynamics simulations are used to investigate microscopic structures and dynamics of methanol and methanol-water binary mixture films confined between hydrophobic infinite parallel graphite plate slits with widths, H, in the range of 7-20 Å at 300 K. The initial geometric densities of the liquids were chosen to be the same as bulk methanol at the same temperature. For the two narrowest slit widths, two smaller initial densities were also considered. For the nano-confined system with H = 7 Å and high pressure, a solid-like hexagonal arrangement of methanol molecules arranged perpendicular to the plates is observed which reflects the closest packing of the molecules and partially mirrors the structure of the underlying graphite structure. At lower pressures and for larger slit widths, in the contact layer, the methanol molecules prefer having the C-O bond oriented parallel to the walls. Layered structures of methanol parallel to the wall were observed, with contact layers and additional numbers of central layers depending on the particular slit width. For methanol-water mixtures, simulations of solutions with different composition were performed between infinite graphite slits with H = 10 and 20 Å at 300 K. For the nanoslit with H = 10 Å, in the solution mixtures, three layers of molecules form, but for all mole fractions of methanol, methanol molecules are excluded from the central fluid layer. In the nanopore with H = 20 Å, more than three fluid layers are formed and methanol concentrations are enhanced near the confining plates walls compared to the average solution stoichiometry. The self-diffusion coefficients of methanol and water molecules in the solution show strong dependence on the solution concentration. The solution mole fractions with minimal diffusivity are the same in confined and non-confined bulk methanol-water mixtures.