Outer measure > $0$? Let's say we have $A \subset I_0$ as an arbitrary set such that
$Int(A) \neq \emptyset$ 
My question is:
is $\mu^* (A)$ always non-negative/positive?

It depends on what kind of measure $\mu^*$ is. 
I assume $\mu^*$ is the outer Lebesgue measure and $I_0 \subset \mathbb R$ an arbitrary interval. Since $\mathring A$ is open and $\mathring A \neq \emptyset$, you find a non-empty open interval $(a,b) \subset \mathring A$. Now we have 
$$ 0 < \underbrace{\mu^*((a,b))}_{=b-a} \leq \mu^*(\mathring A) \leq \mu^*(A) \; .$$ 
