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