$X$ is undecidable but partially decidable.
We have the TM $M$. Just make the state $q$ the final state and make all other final states non-final and get a new TM $M'$. Give input $w$ to $M'$. If $w$ would have taken $M$ to state $q$ (yes case of the problem), our new TM $M'$ would accept it. So, the given problem is partially decidable.
If $M$ goes for an infinite loop and never reaches state $q$ (no case for the problem), $M'$ cannot output anything. This problem is the state entry problem, which like word accepting problem and halting problem is undecidable.