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.

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.

For instance, if $\|\mu\|/\sigma$ is small, then the first moment is $$\frac23\sqrt{\frac2\pi}\frac\mu\sigma+O(\|\mu\|/\sigma)^2.$$

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.)

Therefore, it is very unlikely that a closed-form expression for this moment exists.

As for approximations, those will depend on whether $\|\mu\|/\sigma$ is large or small.

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.