The simplest way to try to understand the idea of identical copies throughout the universe is probably by making a comparison to Minecraft world seeds. In Minecraft, when you create a new world the game first chooses a random number for its seed. Then, as the game generates the world, it uses that seed as the basis for every single choice it makes.
If you had an arbitrarily large number of people playing Minecraft, there are only so many random numbers that can be chosen so there must be duplicates. So some of those people must be playing in a world that is an exact copy of the world someone else is playing.
For the observable universe, there are a very large yet finite number of possible arrangements of the matter within it. You could say that it's similar to how there are a finite number of possible Minecraft seeds, and therefore a finite number of possible Minecraft worlds.
So if the universe is infinite, every single sphere with a radius of ~14 billion light years is effectively a universe independent from all of the others. The paper you reference is basically calculating how many possible seeds there are for a universe, and then based on that number calculating how far away the nearest universe with the same seeds as ours is likely to be.
It is important to understand that 10^(10^29) meters is not a distance, but a radius! If you measured out a sphere with that radius centered on Earth, then it is likely that you will find an exact copy of Earth somewhere in that radius.
With the superpowers you have given your hero, there is no way for him to find a copy of himself. He can't just travel 10^(10^29) meters and expect to find a copy - he has to check every universe from his start to that distance, and in each universe has to check each galaxy to see if it's a copy of the milky way, so he can check if the Earth in that galaxy is a copy of his Earth.
However your hero doesn't have super vision. In order to see a universe he has to be in it. In order to see a galaxy he has to be reasonably near it. The human eye isn't strong enough on its own to see far away galaxies, let alone make out any details about them. He's going to have to jump around a ridiculous number of times to find a universe, then adequately check that universe.
Your hero also doesn't have a super intellect. 10^(10^29) is too large. There are trillions of galaxies in each universe, and they're not aligned to any grid. As he has searches a single universe, he's going to make a number of maneuvers that are not humanly possible to remember. There will not be a straight line from the previous universe through the current universe and to the next universe he will search. Your hero will be hopelessly lost, forever.
It is also not humanly possible for him to remember any significant details about the universes he has already searched. By the time he has searched through a hundred universes, he has seen so many galaxies that the shear amount of information he has seen will have wiped out any previous knowledge he once had. Even if he stumbled across a copy of our universe, he would no longer remember the details of our universe well enough to be able to find the milky way or Earth.
The real nail in the coffin is that without super senses, he can't even reasonably search a single universe. Light takes time to enter the eye, and it has to enter at a normal speed for it to be focused correctly by the cornea. Even if he can effectively instantly travel to wherever he wants to go, he has to stop for the light to reach him and enter his eyes. With more than a trillion galaxies to search and multiple stops to make for each galaxy, he has no hope of ever being able to search enough universes to have a reasonable chance of finding a clone.