My protagonist was a diplomat, and a spy. Things went wrong and he had to flee in a crippled starship. So, he crash-landed on a planet which was inhabit by a primitive life form. He crashed on 16th-century earth.
There is no way to repair his starship with 16th century technology, so he has to stay. However, he is immortal (or, he can expected to survive the next two- to three-thousand years). He wishes to have his starship repaired, however, he is bound by his ethic standards, which ask that he grants humanity their natural development, meaning he can not rush their technical progress for them to build replacement parts for his star ship. He can provide some guidance, but no more than that. This is similar to Star Trek's prime directive, but less strong. He would not be allowed to enslave humanity and force his own culture on them, but he can and will teach them (this includes philosophy as well as science). When he learned this philosophy, he also learned some basics of how societies develop, but he is not a fully trained exo-sociologist.
I think that right after the crash-landing he would take some time to observe the locals.
What would he do then? He needs food, shelter, so, how would he approach the locals? How would he integrate into their society? What would be the topics of his teaching? How would the topics change over the years? My story sets in about 500 years after the crash-landing, so I am looking for a broad overview of what he did in the last fivehundered years, not a detailed discussion of the first few days.
I am sure that he will teach ethics and philosophy. Which other topics?
Initially, I thought about medicine, but he was not a trained doctor in his live in space, and he certainly does not know about the biology of a species he never encountered before. However, he does have alien-highschool, so he knows about bacteria and hygiene.
He is not a scientist, but most highschools and colleges on earth today provide enough of a scientific background to outperform a 16th-century scientist. So alien high-school for sure provided this as well. So maybe he could teach the scientific method and stuff like this?
He has enough of an engineering background to repair his starship, once he acquires the necessary parts. So, could (and would) he teach engineering?
Being a diplomat, he may prove valuable as an advicer to a king or politican, however, if he gets the feeling that he is entagled into politics too much (and thereby influences the development of humanity too much), he will pull back. Maybe this keeps him from assuming such a position in the first place.
He has some martial arts and weapons training. He even learned some military tactics (being a spy may include comanding a division of the special forces). But he will be reluctant to share those skills, because he is pacifistic. He uses force for self defence, maybe even to defend others or prevent crime, but he will stay out of military conflicts, because he would influence the development of humanity too much if he gets involved with those. Also, he will not train an army, especially not if this means that he gives an advantage to one party in a conflict.