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The correct answer is $28$. The given absolute value equation can be rewritten as two linear equations: $4x-4=112$ and $-\left(4x-4\right)=112$, or $4x-4=-112$. Adding $4$ to both sides of the equation $4x-4=112$ results in $4x=116$. Dividing both sides of this equation by $4$ results in $x=29$. Adding $4$ to both sides of the equation $4x-4=-112$ results in $4x=-108$. Dividing both sides of this equation by $4$ results in $x=-27$. Therefore, the two values of $x-1$ are $29-1$, or $28$, and $-27-1$, or $-28$. Thus, the positive value of $x-1$ is $28$.

Alternate approach: The given equation can be rewritten as $\left(4\left(x-1\right)\right)=112$, which is equivalent to $4\left(x-1\right)=112$. Dividing both sides of this equation by $4$ yields $\left(x-1\right)=28$. This equation can be rewritten as two linear equations: $x-1=28$ and $-\left(x-1\right)=28$, or $x-1=-28$. Therefore, the positive value of $x-1$ is $28$.