"in his spare time" or "in my spare time"
Both are idiomatic in English: you can either use “my” or your third person singular pronoun.
The reason for this is because the pronoun comes after “who”. Otherwise you would have to use “my” (for example, “I am his own boss” is ungrammatical).
Here’s an example with his:
I think I am known as a person who expresses his opinion freely about things
Here’s an example with my:
I am also a person who buries my emotions
(Both these examples were found with COCA using collocation searches.)