Mathematica cannot compute the first moment of the projection, even for $\sigma=1$:
(In fact, by rescaling, the value of $\sigma>0$ does not matter here.)
Similarly, Mathematica cannot compute the second moment of the projection.
Therefore, it is very unlikely that a closed-form expression for either one of these moments exists.
As for approximations, those will depend on whether $\|\mu\|/\sigma$ is large or small.