The patient has loss of pain and temperature on the right side of his face. Sensation of the face is ipsilateral, so the issue must be on the patient's right side, which we can confirm by knowing that sensation of the body is contralateral, and he has lost left sided pain and temperature of the body.

Pain and temperature sensation of the body is part of the spinothalamic tract, which always runs laterally through the brainstem. This can be confirmed by remembering that sensation to the face also runs laterally through the brainstem. So, we can confirm this is a right sided lateral brainstem issue.

The loss of gag reflex and paralysis of the vocal cords imply impairment of cranial nerves IX and X, both of which localize to the medulla. Therefore, the answer is right dorsolateral medulla.

Lateral Medullary Syndrome (Wallenberg) - see FA for more details. Symptoms for this syndrome match up with what is being presented.

I get why its lateral but dont all cranial nerve except 4 arise Ventrally? WTF they add this Dorso before lateral?

This was lateral medullar syndrome aka dorsolateral infarct of the PICA.