Which version is grammatically correct: "She was admitted to practice law in 2001" or "She was admitted to practice law on 2001"?

The difficulty here is the ambiguity of prepositions in English.  “In” generally refers to the question “where?” (Where is the cinnamon?  In the cupboard.”)  “On” also usually refers to the question of “where?” (Where is the cinnamon?  On the shelf.”)  These prepositional phrases both refer to a physical location.  But in your example, “in” and “on” both refer to a time of occurrence (When is your 75th birthday?  In 2017.) (When is your birthday? On the 24th of November.)  So your choice depends on whether the object of the preposition is exact (She was admitted…on Jan. 1, 2001) or within a range of times (She was admitted… sometime in 2001).  So in your example, “She was admitted to practice law in 2001” is correct if the speaker is referring to the first time she was admitted.  There could be a case where the speaker is referring to a past event:  “While she is not permitted now to practice law, she was permitted to do so in 2001, a privilege since revoked.” Better grammar would refer to a more exact time:  She was admitted to practice law on this exact date:  January 1, 2001.”