No, but he did die so many times that he could no longer remain among the living, even though King Yama granted him another life. He instead traveled with Shenron to the Other World to continue his existance until the day Earth would need his help again.
Sorry, but that's frankly absolute bullshit, you need to watch the show a lot more before you can reply to these questions. The reason Goku had to go to Otherworld was because the Dragon Balls had been used so many times so Goku sacrificed his life to make one final wish and restore balance to the balls. Goku died twice, whereas many other characters died three times, such as Krillin and Chiaotzu, therefore Goku had not died so many times that he had to return to Other World. To answer the question no, Goku does not die of old age, as already made apparent.
He died so many times and brought back but by the end he becomes guardian of the Earth enabling him to live forever.
And he has another 1000 years of life left, due to Elder kai giving him his life force. He left so the earth could remain safe