How to practically find solutions to a discrete logarithm?
256-bit discrete logarithms on a prime field are definitely not of the order of magnitude used in cryptographic applications. Secure sizes for this problem are in the thousands of bits, very much like integer factorization.

To break that example discrete logarithm, you probably want to use Index Calculus, more specifically the Linear Sieve. Resorting to the Number Field Sieve is probably overkill for such a small prime field, and GGNFS is not prepared to solve discrete logarithms, although much of it could be reused to do so.

Note that 256-bit sizes are OK for problems with less structure, where only generic discrete logarithm algorithms, like Pollard's Rho, work --- such as elliptic curves over a 256-bit prime.