What is the right way to extract particular frequency phase information from FFT?
To estimate spectral component phase angle relative to shaft angle using an FFT of length N, first gather a sequence of N samples such that the shaft angle is known with accuracy at sample number N/2, then window with a Nuttal window and do an FFT shift before the FFT.  The  atan2() of the FFT bin result closest to the desired spectral component will produce an estimate of the phase relative to the center of the original sample window.  Adjust that phase by the known shaft angle at that N/2 sample point to get a spectral phase relative to shaft angle.

If you know the spectral frequency or use a frequency peak estimator, then the phase can also be interpolated between bins for increases phase estimation accuracy.  Interpolate phase by using a Sinc interpolation seperately on the real (even) and imaginary (odd) FFT result components before taking the arctangent.