What is the right way to calculate the inverse Z-transform of $zX(z^{-1})$ say the signal $x(n)$ has the z transform $X(z)$ and there is signal $x_1(n)$ that 

$X_1(z)=zX(z^{-1})$

I tried 2 different approach to get the relationship between $x(n)$ and $x_1(n)$ and the result is different without any error being noticed

Approach 1:

$x_i(n)=x(-n) \rightarrow X_i(z) = X(z^{-1})$

$x_1(n)=x_i(n+1) = x(-n-1) \rightarrow X_1(z) = zX_i(z)=zX(z^{-1})$

Approach 2:

$x_i(n)=x(n-1) \rightarrow X_i(z) = z^{-1}X(z)$

$x_1(n)=x_i(-n)=x(1-n) \rightarrow X_1(z) = X_i(z^{-1}) = zX(z^{-1})$

Hence in the end, I don't know $x_1(n) = x(-n-1)$ or $x_1(n) =x(1-n)$ and I can't find any mistake. I know that different function can have same z-transform but different ROC but in this case, no any info of ROC is given
The first result is correct, the second is wrong. You're right that $x_i[n]=x[n-1]$ corresponds to $z^{-1}X(z)$. And if you replace $z$ by $1/z$ you need to replace $n$ by $-n$. But $x_i[-n]$ is simply $x[-n-1]$,  and not $x[-n+1]$. Note that you don't replace the whole argument of $x[n-1]$ by its negative value, but you just replace $n$ by $-n$.