How to calculate arbitrary phase shift in discrete signal? When I have digital audio signal, represented in samples, I can easily phase-shift the signal by 180° by simply inverting the value of each sample.

But what if I want to phase shift an arbitrary voice audio signal by say +90° or -90°?

How do I transform an arbitrary, discrete signal by phase-shifting it by an arbitrary phase angle?
The answers to this question explain the solution for the continuous-time case. In discrete time, the solution is completely analogous. The impulse response of a phase shifter with phase shift $\theta$ is given by

$$h[n]=\cos(\theta)\delta[n]+\sin(\theta)g[n]\tag{1}$$

where $g[n]$ is the impulse response of an (ideal) discrete-time Hilbert transformer. Eq. $(1)$ shows that the solution is just a linear combination of the identity system and a Hilbert transformer (which performs a $90$ degree phase shift).