Well first off, it's not a paradox. It's an illustration. Specifically, it's an illustration of:
Cat is a macro object. He can be only in 2 states
Macroscopic objects like cats have distinct states like "live" or "dead". Microscopic objects, like quantum observables, do not have two states, they can exist in a mixture. People did not like this, famously Einstein, and many simply threw up their hands and said 'well it's only microscopic so who cares!"
Schrodinger's cat is an attempt to show that that last statement is wrong.
The box contains a decaying atom which we knew beyond hope at that point really could be both decayed and not decayed at the same time. We call this "superposition". I should point out that most "observables" have much more than two states, like "position" which is basically continuous, but he deliberately chose one with two distinct states to make it simpler to understand.
Ok, so due to the setup, if the atom is decayed then the cat is dead, and if its not, he's alive. But the atom isn't either of those things, its both. So then what state is the cat in?
the fact that you don't have the information about his state
doesn't mean he is in both states at ones.
That's exactly what it means.
I prefer to think of it in slightly different terms, I say there is a 50% real dead cat and a 50% real live cat. It's only when you collapse the state of that particle that one of those two becomes 100% real and the other 0% real. It is the purpose of the thought experiment to show that the weirdness "leaks out" of the microscopic world.
Now the pedants among the physics world generally dismiss this thought experiment with the wave of a hand. They tend to invoke a bit of magic called decoherence and suggest that you don't need to open the box because it's interactions with the real world essentially do that for you. You might not be looking inside, but certainly the photons from the sun are.
The problem is that it is possible to construct real-world experiments where the collapse can be isolated enough to end up with this same result. And this is the thing that really freaks people out, we still have no idea what "collapse" is, but it's core to the entire concept. There's various suggestions, and theories with no collapse, but none seem terribly convincing.
Now if you want to get really freaked out, what if there is someone watching you carry out the experiment? Does the original particle only collapse when they look at you? So does that mean you're only 50% real? That quickly leads to madness...
The sun in the Alpha-Centauri system has defined properties whether
or not you have the information about its state
Nope, not at all. And this has been demonstrated in the lab, repeatedly. This is what Aspect's experiment showed, for instance, by changing the measurements in mid-flight of the particles.