He plays his MP3 player in the library. I don't appreciate that.
While not being sure, I just guess it should be like below:
I do not appreciate his playing his MP3 player in the library.
He plays his MP3 player in the library. I don't appreciate that.
While not being sure, I just guess it should be like below:
I do not appreciate his playing his MP3 player in the library.
The sentence
I do not appreciate his playing his MP3 player in the library.
is grammatically correct, although the first use of the possessive pronoun his may be considered out-of-place if the sentence is spoken rather than written (see these related questions). Especially in modern spoken English, the following sentence may sound more natural:
I do not appreciate him playing his MP3 player in the library.