What's the mistake in the annihilating polynomial?

For the matrix

$$A=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$$

an annihilating polynomial is $x^3-1$, but its characteristic polynomial is $x^3+1$.

Is this a contradiction, since roots of characteristics polynomial should be roots of annihilating polynomial too?

You just made a mistake calculating the characteristic polynomial. The characteristic polynomial is $-x^3 + 1$, which is just your annihilating polynomial times $-1$.
To find the characteristic polynomial, we need to find $\det(A-\lambda I)$, which is