Orthogonal projection of $v = (0,0,5,0)^T$ onto the subspace $W$.

To find the orthogonal projection of $$v = (0,0,5,0)^T$$ onto the subspace $$W$$ spanned by $$(-1,-1,-1,1)^T,(-1,1,1,1)^T,(-1,-1,1,-1)^T$$.

Since the vectors which forms a basis for $$W$$ are already orthogonal, the orthogonal projection of $$v = (0,0,5,0)^T$$ onto the subspace $$W$$ is the sum of the projection of $$v$$ on each of the basis vectors.

Is my logic and method correct?

Yes, it's fine assuming that by projection of $$v$$ on a a vector $$w$$ you mean $$\frac{\langle v,w\rangle}{\lVert w\rVert|^2}w$$.