Let us fix your notation first, there are some mistakes: an operator $\hat{A}$ acts on a state $|\psi\rangle$ of a Hilbert space, not on a wave function. The wave function $\psi(x)$ is the scalar product $\langle x|\psi\rangle$ of a state against the position basis. As such, your notation
$
\hat{x}\psi = x_0\psi
$
is wrong if by $\psi$ you assume the wave function. It should rather be
$$
\hat{x}|\psi\rangle=\hat{x}\int dx'|x'\rangle \langle x'|\psi\rangle = \int dx'x'\psi(x')|x'\rangle
$$
if we expand $|\psi\rangle$ onto the position basis. If you want $|\psi\rangle$ to be an eigenstate of the position operator then one can show that the kernel of the above integral does not belong to the collection of summable functions (with respect to the $L^2$-norm on the real line): formally however the above equation is satisfied if one takes $\psi(x') = \delta(x-x')$, although this is not a physically realisable state (unless one construct physical states as a combination therefrom). However, the position operator is an unfortunate example; this said, the above has nothing to do with the collapse of the wave function.
The collapse of the wave function
Let $|\psi\rangle$ be a state as element of a Hilbert space and let $\hat{A}$ be an observable, namely a self adjoint operator on that Hilbert space such that the initial state lies in the domain thereof. By definition, then, $|\psi\rangle$ can be expanded onto a basis of $\hat{A}$, namely one can write
$$
|\psi\rangle = \sum_a c_a|a\rangle
$$
with $|a\rangle$ being the collection of eigenstates of $\hat{A}$ and $c_a$ some ($L^2$-summable) coefficients. After a measurement (experiment) of the observable $\hat{A}$ on the state $|\psi\rangle$ the aforementioned state collapses into one of the eigenstates of $\hat{A}$, namely
$$
|\psi\rangle \to |a\rangle
$$
producing, as such, the eigenvalue $a$ as result of the measurement process. If nothing else happens (no evolution), then the state lies into an eigenstates indefinitely, as the eigenstate of an operator cannot be expanded into a combination of the other eigenstates (because they are linearly independent). If some other things occur, for example time evolution or interactions, then the eigenstate $|a\rangle$ can change in turn to become some other state $|\phi\rangle$.
If we make another measurement of the observable $\hat{A}$ then the state can collapse into any other of the eigenstates, it needs not always collapse into the same one. After having performed infinite measurements, each of the eigenstates $|a\rangle$ might have occurred with a probability distribution given by (the square of) the coefficients $|c_a|^2$.
