According to the minhag of the Ari not to cut hair until Shavuot – does this also apply to not listening to music till Shavuot.
No. The idea of not cutting hair at all during the entire Omer period is related (based on kabbalah) specifically to hair, and is unrelated to the mourning practices of the Omer, such as listening to music.
Note that according to some, the custom of not listening to music applies even after lag ba’omer, irrespective of the Arizal. Many, however, are lenient in the second half of the Omer.