If $A\in \mathbb{C}^{n\times n}$ with eigenvalues $(\mu_1,\ldots,\mu_n)$, is there anything we can say about the eigenvalues of $T = \Re(A)$; let's call them $(\lambda_1,\ldots, \lambda_n)$? Especially, does it hold that $|\lambda_i|\leq|\mu_i|$?
The last part holds for the largest eigenvalue via a norm argument, but I can't really come up with anything for the other eigenvalues.
No that is not true in general. Take for example $$\begin{bmatrix}1 & -i\\i & 1\end{bmatrix} $$
This has eigenvalues $0$ and $2$. The real part is just the identity and has repeated eigenvalue 1.