SHA-256 Reversing A String of Equal Length
SHA256 gives 256 bits, which is 32 bytes, not 64 "characters" (please use the well-defined bits and bytes). When written in hexadecimal notation, you need 64 characters from 0-9a-f, but the length of the hash is 256 bits. 

Yes, you could invert the hash. Unfortunately, a straightforward lookup table would be too large to store, but you can still do brute force without memory. This would take you on average $2^{255}$ tries (that's many). 

There have been papers, like "Bicliques for Preimages: Attacks on Skein-512 and the SHA-2 family" which discuss preimage resistance attacks on the SHA-2 family, but they're, as far as I know, all academical breaks (meaning they cannot really be exploited efficiently.