You can get the maximum principle for subharmonic functions: Write $F$ for the supremum of $f$ in $\Omega$, write $A$ for the set where $f = F$ and $B$ for the set where $f < F$. Then
$\Omega = A \cup B$.
Now, B is open by upper semicontinuity. If the first is non-empty then it is fairly straightforward to show that it too must be open and therefore the whole domain.