Does the mean value change? 
We know that $x_0$ and $p_0$ are the mean values for the position and momentum of a particle in the normalized state characterized by the function $\psi (x)$, ( that is, $x_0=\langle x \rangle_\psi$ and $p_0=\langle p \rangle_\psi$).
Does the mean value of $x$ change for the function $\psi(x+x_0)$ ?

If the mean value for x for the funtion $\psi(x)$ is:
$$\langle x \rangle_\psi=\int_{-\infty}^\infty \psi^\ast (x)\;  x\; \psi(x) dx=x_0$$
The mean value for x for the function $\psi(x+x_0)$ is:
$$\langle x \rangle_{\psi' }=\int_{-\infty}^\infty \psi^\ast(x+x_0)\; x\; \psi(x+x_0) dx$$
If we center the function in $x=x_0$, we have:
$$\langle x \rangle_{\psi' }=\int_{-\infty}^\infty \psi^\ast(x)\; (x-x_0)\; \psi(x) dx$$
Then
$$\langle x \rangle_{\psi' }=\int_{-\infty}^\infty \psi^\ast(x)\; x\;\psi(x)dx -\int_{-\infty}^\infty \psi^\ast(x)\;x_0\; \psi(x) dx$$
Where the first integral is $\langle x \rangle_\psi$ and the second one, we know that
$$\int_{-\infty}^\infty \psi^\ast(x)\psi(x) dx=1$$
Because it is normalized, so:
$$\langle x \rangle_{\psi' }= \langle x \rangle_\psi -x_0$$
Where $\langle x \rangle_\psi=x_0$, then:
$$\langle x \rangle_{\psi' }=0$$

Is that correct?
If so, what will be the mean value for $p$ with $\psi(x+x_0)$?

 A: Your result is correct. You have a function $\psi(x)$ whose mean is $x_0$. So the function $\psi(x+x_0)$ is the original function shifted by an amount $|x_0|$ towards $x=0$. But then this means (pun always intended) that your new mean has to be at $x=0$. As others have pointed out, this only works for an integral from $-\infty$ to $\infty$. In general if you are finding the average of something over a finite interval, you would have to shift your interval as well (there are some contrived exceptions to this, but I won't go into them here). 
As for the average momentum, I would think nothing changes as long as your system has translational invariance, but @kryomaxim seems to think it does matter in general. I don't think the correct momentum operator would involve $\frac{\partial}{\partial(x+x_0)}$, because you are still working in the original position basis. Many arguments in QM text books I have seen exploit the fact that we can center the wavefunction so that $\langle X\rangle=0$ without changing the mean momentum. So I believe shifting the wavefunction to a new position will not change the mean momentum if there is translational invariance in the system.
A: If you have an integral over the interval $[-\infty, \infty]$, yes, that's correct.
The wave function $\psi'$ is simply displaced by $-x_0$, so the Position expectation value does Change exactly this amount. For finite intervals, note that also the Integration endpoints Change.
For computing the expectation value $p$, use the Definition of the Momentum Operator and use the fact that $\frac{\partial}{\partial x} = \frac{\partial}{\partial (x+x_0)}$. 
