I would phrase it as "He may look rude, but he is very kind on the inside. Though from your choices internally would be the better one.
"From inside" is a direction from inside to outside. So for a sentence like "They heard a sound from inside the barrel" It's the sound that moves from inside the barrel to outside (where the listener is).
"Internally" is for something that happens inside, and stays inside. So here you would have a sentence like "The group took care of the problem internally". Here whatever the group did, the outside will never know.
Same goes for the reverse of from outside and externally, just opposite of course :)