Mathematica cannot compute the first moment of the projection, even for $\sigma=1$: 

[![enter image description here][1]][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. 

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


  [1]: https://i.sstatic.net/EiY9K.png