The correct prepositional usage is: "begged pardon from someone".
This structure expresses that someone is seeking forgiveness directly \textit{from} another person.
Let's examine each option:
(A) \textit{"begged pardon of"} – This is archaic or incorrect in modern usage.
(B) \textit{"begged pardon with"} – Incorrect; the preposition "with" implies partnership, which does not fit the context.
(C) \textit{"begged pardon from"} – Correct; shows the act of asking for forgiveness from his teacher.
(D) \textit{"begged pardon for"} – Incorrect; "for his teacher" implies he is asking forgiveness on behalf of his teacher, which is not intended.
Therefore, the most grammatically accurate and contextually correct phrase is: "He begged pardon from his teacher."