Yes, the objective, personal pronoun 'him' functioning as the object of the preposition 'at' is the correct form.
However, the preposition 'at' is a bit clumsy. A more appropriate preposition in this sentence is 'for'.
Example: Both the teachers and the students were delighted for him receiving such high scores.
This is easier to see when the sentence is simplified, "Teachers and students were delighted for him."
The pronoun in the sentence, 'him', is correct if the one the sentence refers to is a male. The pronoun 'him' is a singular, objective, personal pronoun which is functioning as the object of the preposition 'at'.The preposition 'at' is not the best choice, a better choice is 'for' (were delighted for him).Note: The word 'both' can function as an indefinite pronoun when it takes the place of a noun. However, in this sentence, it functions as an adjective, describing the nouns 'teachers and students'.Example use as a pronoun: The teachers and the students were both delighted...
The noun in the sentence is "students" and "teachers" as they refer to people.
It's not clear - I think you mean, "Either the students or the teachers can join." Or perhaps you mean, "Both students and teachers can join."
Teachers expect compliance from their students.
A good teacher is a person who helps his students want to learn.
no teachers don't tickle students
Some teachers use an authoritarian style of discipline to make the students behave
Parents' abdication of their responsibility is an issue for teachers trying to help problem students.
why should teachers challenge students
The teachers' classroom was decorated for the holidays.
Teacher In Service Days are days when the teachers have the day to themselves without any students to bother them.
Students would need to seek asylum if teachers knew how much work they plagiarised.