No, if you were traveling at the speed of light, you would not be able to hear sounds before they are made. According to our current understanding of physics, as an object with mass approaches the speed of light, its relative time slows down. This phenomenon is described in Einstein's theory of relativity.
When an object travels at or near the speed of light, time dilation occurs, which means that time appears to pass more slowly for the moving object compared to an observer at rest. This effect is significant as the object's speed approaches the speed of light, but it is not possible for an object with mass to reach or exceed the speed of light itself.
Since sound requires a medium, such as air or another substance, to propagate, it cannot travel at or faster than the speed of light. In other words, light travels much faster than sound. So even if you were traveling at nearly the speed of light, sound waves would not catch up to you before they were produced.