Most Helpful Girl
Hypothetically, yes, because the headphones will be stationary relative to him. The sound coming from them wouldn't be interfered with as a result of the speed he's moving at because it goes basically directly to his ear drums.
Realistically, we'll never know for sure. Things travelling at the speed of light experience an increase in mass as well. Hell, the rate at which their mass increases will go beyond infinity. As would the amount of energy required to reach this speed so.. one can dream.1
Most Helpful Guy
No, because he is no longer a jogger. He is then a freak of nature. But, maybe yes? Because he is going at the same speed as the sound, so it might be a little slower than normal, but if he wasn't deaf from the sheer speed he was going, possibly.1