If you have upper and lower bounds M and 0, then you can apply [Popovivicu's upper bound on variance][1]:
$$\sigma^2<\frac 1 4(M-m)^2$$

Once you have the variance, apply usual sample mean distribution logic, i.e. the variance of a sample mean $\bar x$ to be $\sigma^2_{\bar x}\sim\sigma^2/n$. The rest is trivial.

  [1]: https://en.wikipedia.org/wiki/Popoviciu%27s_inequality_on_variances