Is it feasible to build an index of prime factors? Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing prime keys up to $2^{2048}$?

I understand this would take an immense amount of time to do, but it would only be done once. Obviously, computing such an index would take a considerable amount of time and would have to be done with someone having the right amount of resource. Given the public key, you would be able to look up what the two primes are, and hence retrieve the private key instantly.

Breaking an 256 AES would then be easier, as AES keys are often generated using the RSA private key.

Would it be possible to build such an index? How long would that take in permutations and perhaps with the fastest computer in the world?
Let's assume for an instant that you could build a large table of all primes. Then... what ? How would you use it ? What would you look up ?

If you "just" scan the table and try to divide the number to factor by each prime, then this is known as trial division; there is no need to store the primes (they can be regenerated on-the-fly; that's the division which is expensive). Yet this is an highly inefficient factorization algorithm. We know much better factorization algorithms (which nonetheless utterly fail at making the slightest scratch on the surface of a 2048-bit RSA modulus).

Of course, the list of 1024-bit primes is so large that storing it, or even simply generating each prime, is ludicrously infeasible. There are about $2^{1014}$ such primes; that's close to  $10^{308}$. Suppose you are an omnipotent but severely bored deity, and you decide to store these primes, using a storage device which uses the size of an hydrogen atom for each prime (we, as humans, can certainly not store that much information in so small a space, but hey, that's a piece of cake for an omnipotent God). The known universe has an approximate volume of $10^{79}$ m3. The volume of an hydrogen atom is close to $10^{-30}$ m3. So our eccentric divinity can pack about $10^{109}$ values in the whole Universe. He would still need $10^{199}$ complete Universes to store them all. $10^{199}$ is a mind-boggling number (if your mind is not boggled by it, then it must already be much more boggled by something else). It is ten billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions of billions. In other words, a lot. And we are still talking about complete Universes packed with atom-sized integers.