If he's an orthodox fighter, when you see that arm twitch, disregard what he's throwing and slam your leg kick into his gut OR throw your right leg into his leg. Unless you're fighting Inspector Gadget, I imagine that your legs should be long enough. For good measure to shut down his punch, either put your glove on the side of his head to pull him INTO the body kick and then step through to the outside.
If he's Southpaw so the exact same thing but through your right leg into the body and left leg into his leg. Which stance you're in won't make a meaningful difference, once you're practised at it.
Because he's specifically a puncher you need to close the door, so bring yourself into the long guard after each combination you throw as coverage.
Here's an exercise to help you find a strong blocking position.
Put one hand on the side of your face, and cover half of it (like you're trying to be the phantom of the opera) slide your hand up until you can reach and pull the back of your head, you should feel that side of your head COMPLETELY guarded with the elbow pointing outwards. Now good luck to him, if he jabs, he'll jab your elbow, if he hooks, you'll block it.
When you feel the hook, loop your arm round and entangle his arm, and make sure that while you have his arm looped, the back of your fist is pressed against his chest. Get a strong lock with to the other hand round the back of his neck. You have now got either a tie up in boxing or kickboxing that the referee will have to break - or you have got into a clinch that you can fight from.
