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###Edit

(Corrected error in algorithm)

You can reverse the encryption with Solve, which can handle modular equations.

With[{alphabet = Alphabet[]},
  With[{n = Length[alphabet]},
    EncryptChar[char_, a_, b_] :=
      Module[{position, shift, mod},
        position = Position[alphabet, char][[1, 1]] - 1;
        shift = Mod[a position + b, n];
        alphabet[[shift + 1]]];
    DecryptChar[char_, a_, b_] :=
      Module[{position, shift, mod},
        position = Position[alphabet, char][[1, 1]] - 1;
        shift = Solve[a x + b == position, x, Modulus -> n][[1, 1, 2]];
        alphabet[[shift + 1]]]]]

(DecryptChar[EncryptChar[#, 3, 5], 3, 5] & /@ Alphabet[]) == Alphabet[]

True

Note: I also cleaned up you code a bit.