I've been struggling with this question for some time. Here is what I have so far:
I guessed that the inverse of $T - \lambda I$ is of the form $c(T+\lambda I)$. Then, solving for $I = (T-\lambda I)c(T+\lambda I)$, we get the equation $c = \frac{-1}{1 + \lambda^2}$, which is well-defined for $\lambda \not= \pm i$.
Thus, the spectrum of $T$ only contains $-i$ and $i$. (It has to contain these two since the spectrum is non-empty and closed). But I have no clue how to figure out if $i$ and $-i$ are in the point or continuous spectrum. (They're not in the residual spectrum since the residual spectrum of unitary operators is empty and $T$ is unitary). Any ideas would be helpful.
Point spectrum.
Since $T$ is normal, the spectral theorem applies. Let $P_+$ and $P_-$ be the spectral projections on $\{+ i\}$ and $\{-i\}$ respectively. These are both nontrivial since $T \ne \pm i I$. Then $T P_+ = i P_+$ and $T P_- = -i P_-$, and nonzero vectors in the range of $P_\pm$ are eigenvectors for $\pm i$.
If you don't want to use the spectral theorem, you can do it explicitly: $P_+ = (I - i T)/2$ and $P_- = (I + iT)/2$. Show that these are orthogonal projections with $T P_\pm = \pm i P_\pm$.