The partition function $Z$ is defined as
$$Z(T,\mathbf{x})=\sum_{\{\mu\}}e^{-\beta \mathcal{H}(\mu)}$$
where sum is over microstate ($\mu$).
For an ideal gas,
$$Z(T,V,N)=\int \frac{1}{N!} \prod_{i=1}^N\frac{d^3q_id^3p_i}{h^3}\exp\left[-\beta \sum_{i=1}^N\frac{p_i^2}{2m}\right]$$
$$Z(T,V,N)=\frac{1}{N!}\left(\frac{V}{\lambda(T)^3}\right)^N$$
$$\Rightarrow \langle H\rangle =-\frac{d\ln Z}{d\beta}=\frac{3}{2}\frac{N}{\beta}$$
$$\langle H^2\rangle_c=-\frac{\partial \langle H\rangle }{d\beta}=\frac{3}{2}\frac{N}{\beta^2}$$
$$\sqrt{\langle H^2\rangle_c}\propto \sqrt{N}k_BT$$
As proposed.
Further In general,
$$\langle H^2\rangle_c=-\frac{\partial \langle H\rangle }{d\beta}=k_BT^2\left.\frac{\partial \langle H\rangle }{\partial T}\right|_\mathbf{x}$$
$$\Rightarrow \langle H^2\rangle_c=k_BT^2C_\mathbf{x}$$
The above is true in general for canonical ensemble.