According to our current understanding of physics, it would not be possible for someone to travel to the other side of the observable universe within their own lifetime, even at speeds close to the speed of light. The vast scale of the universe, combined with the limitations imposed by the theory of relativity, make such a journey unfeasible.
As an object with mass approaches the speed of light, its relativistic mass increases, and time dilation occurs. This means that as an object accelerates to higher speeds, time slows down for that object relative to an observer at rest. From the perspective of the traveler, their journey may subjectively take a shorter amount of time due to time dilation, but from an external observer's standpoint, the journey would still span millions of years.
Moreover, the observable universe has a finite size. The current estimate for the observable universe's diameter is about 93 billion light-years. This means that the light from the farthest regions of the universe, which we can observe, has taken about 13.8 billion years to reach us. Even if someone were to travel at speeds close to the speed of light, they would still be limited by the speed of light itself and could not surpass it. Therefore, reaching the other side of the observable universe within a human lifetime would not be possible.
It's worth noting that our understanding of the universe is continually evolving, and future scientific advancements may lead to new insights. However, based on our current knowledge, traveling to the other side of the universe within a single lifetime is beyond the realm of possibility.