Starting with the first question
What can be said about how long it takes for the underlying stochastic process to deviate by a given amount $\delta x$?
we can say something about the variance of the process.
As I have previously answered in another related question on the physics site, the variance of a process after time $\tau$ is
$$\langle x(\tau)^2 \rangle = \tau^2 \int_0^\infty S_{\dot{x}}(\omega) \left( \frac{\sin(\omega \tau / 2)}{\omega \tau / 2} \right)^2 \frac{d \omega}{2\pi}$$
where $S_{\dot{x}}(\omega)=\omega^2S_x(\omega)$ is the spectral density of the velocity of the random process$^{[a]}$.
Now to the second question
What can be said about the probability distribution of the process at a time $t$ after it is measured to be at say $x_0$?
You're basically asking for the conditional probability $P(x, t| x_0, 0)$, i.e. given that the process was at $x_0$ at time $0$, what is the probability distribution of the process at time $t$?
That question cannot be answered from the spectral density alone.
As Marcus Muller's answer explains, the spectral density is equivalent to the correlation function, which is less information than the full conditional probability.
However, with some other assumptions, more can be said.
For example, if a process is both Markov and Gaussian, then, as proven by Doob, the conditional probability is given by$^{[b]}$
$$P(x, t| x_0, 0) = \frac{1}{\sqrt{2 \pi \sigma_t^2}}\exp \left( - \frac{(x - \mu_t)^2}{2 \sigma_t^2}\right)$$
where
\begin{align}
\mu_t &= \mu + \left(x_0 - \mu \right) \exp(-t / \tau)\\
\sigma_t^2 &= (1 - \exp(-2 t / \tau))\sigma^2 \, .
\end{align}
Here $\mu$ and $\sigma$ are the mean and standard deviation of the process and $\tau$ is the relaxation time.
In fact this theorem guarantees that a Gaussian and Markov process is completely defined by those three parameters.
Of course without the Gaussian and Markov assumptions, much less can be said.
You could ask, for example, what is the mean time it takes for a process to fluctuate by an amount $dx$ for the first time.
That is called "mean first passage time", and we cannot find it from just spectral density.
For whatever it's worth, one can reasonably easily compute mean first passage time for a discrete time random walk.
In fact, there is a delightfully eclectic set of solutions to that problem in this Puzzling Stack Exchange post.
There's probably some way to do it for continuous processes too, but right now I do not know how.
Perhaps in the Gaussian and Markov case it can be solved...
$[a]$: We're assuming that the process is stationary.
$[b]$: See this reference (pdf).