For i.i.d. random variables $X_1, \dotsc, X_n$, the unbiased estimator for the variance $s^2$ (the one with denominator $n-1$) has variance:
$$\mathrm{Var}(s^2) = \sigma^4 \left(\frac{2}{n-1} + \frac{\kappa}{n}\right)$$
where $\kappa$ is the excess kurtosis of the distribution (reference: Wikipedia). So now you need to estimate the kurtosis of your distribution as well. You can use a quantity sometimes described as $\gamma_2$ (also from Wikipedia):
$$\gamma_2 = \frac{\mu_4}{\sigma_4} - 3$$
I would assume that if you use $s$ as an estimate for $\sigma$ and $\gamma_2$ as an estimate for $\kappa$, that you get a reasonable estimate for $\mathrm{Var}(s^2)$, although I don't see a guarantee that it is unbiased. See if it matches with the variance among the subsets of your 500 data points reasonably, and if it does don't worry about it anymore :)
The distributions of the sample mean and variance of a normal distribution are well-known (normal for the mean, Chi square for the variance). As whuber says, you can't find the pdfs of the sample mean $\overline{x}$ and, especially, the variance $s^2$ except in special situations. Given only the population mean $\mu$ and variance $\sigma^2$ and nothing else, all you can find exactly are sample mean and variance of $\overline{x}$ and the mean of $s^2$ (but not $s$):
Let the sample size be $n$. Then every introductory text on statistical theory demonstrates that:
$$
E(\overline{x})= \mu
$$
$$
Var(\overline{x}) = \frac{\sigma^2}{n}
$$
and
$$
E(s^2) = \sigma^2
$$
If you know, in addition to $\mu$ and $\sigma^2$, the population fourth central moment $ \mu_4 = E[(X =\mu)^4]$, you can also compute the exact variance of $s^2$
$$Var(s^2) = \frac{(n-1)^2}{n^3}\left(\mu_4 - \frac{n-3}{n-1}\sigma^2\right)
$$
Reference:
CR Rao (1965) Linear Statistical inference and its applications, Wiley, New York, p.368.
Best Answer
If the data are normal, the sample mean and the sample variance are independent. If the data are not normal, the covariance/correlation between the sample mean and the sample variance are $O( \kappa_3 n^{-1})$ where $\kappa_3$ is the (central) skewness of the distribution. If, for any reason, you really need the sample mean and the sample variance to be independent, then you can calculate these statistics on independent subsets of data. However, as other people noted, you will suffer efficiency losses.
Cross-validation that you've been thinking of, essentially, fights correlations between different statistics computed on the same data. There are applications where that's a dire necessity. E.g., in regression, the correlation between $\hat y_i$ and $y_i$ is given by the hat-value $h_{ii}$, the diagonal entry of the projector matrix $X(X'X)^{-1}X'$, and is $O(1)$, so you may need to suppress it when you talk about model selection or residual diagnostics (or at least control for it with degrees of freedom corrections like $n-p = n-\sum_i h_{ii}$). But there are applications where splitting the sample is a ridiculous overkill.