We present the results of experimental and theoretical studies of fast anisotropic relaxation and rotational diffusion in the first electron excited state of biological coenzyme NADH in water-ethanol solutions. The experiments have been carried out by means of a novel polarization-modulation transient method and fluorescence polarization spectroscopy. For interpretation of the experimental results a model of the anisotropic relaxation in terms of scalar and vector properties of transition dipole moments has been developed based on the Born-Oppenheimer approximation. This model allows for the description of fast isotropic and anisotropic excited state relaxation under excitation of molecules by ultrafast laser pulses in transient absorption and upconversion experiments. The results obtained suggest that the dynamics of anisotropic rovibronic relaxation in NADH under excitation with 100 fs pump laser pulses can be characterised by a single vibrational relaxation time τv lying in the range of 2-15 ps and a single rotation diffusion time τr lying in the range of 100-450 ps, both depending on ethanol concentration. The dependence of the times τv and τr on the solution polarity (static permittivity) and viscosity has been determined and analyzed. Limiting values of the term P2(cos θ) describing the rotation of the transition dipole moment in the course of vibrational relaxation have been determined from experiments as a function of ethanol concentration and analyzed.