Can we map SHA-256 output bits to fixed-length input bits?
No this is not possible.

Basically SHA-2 consists of many rounds where the input is confused using irreversible functions. Each time such an irreversible function is called another variable is introduced. Furthermore these functions are not operating on a single input/output bit (of course) - they are dependent on all the input bits due to the inherent transposition.

So the reverse function will consist of many many variables, that cannot be simplified to a function without one. Any SAT solver will happily break its teeth on it. And that's why you cannot just create a generalized mapping function.

For attacks usually a reduced number of rounds are used, trying to expand them to the full hash later. And if your input $N$ is small enough then you could simply use brute force or an attack using a rainbow table... but that's about it.