Estimating Filtered Noise Variance
I assume that you have a wide-sense stationary discrete-time input noise signal $x(n)$, a linear time-invariant filter with impulse response $h(n)$, and an output noise process $y(n)$. If we assume that $x(n)$ is zero mean (which is not necessary, but makes it easier) and if we model $x(n)$ as perfectly band-limited, we get for its variance

$$\sigma_x^2=\frac{1}{2\pi}\int_{-\pi}^{\pi}S_x(e^{j\theta})d\theta=
\frac{1}{\pi}\int_{\theta_1}^{\theta_2}S_x(e^{j\theta})d\theta\tag{1}$$

where $S_x(e^{j\theta})$ is the power spectral density of $x(n)$, $\theta$ is the normalized frequency $\theta=2\pi f/f_s$, and $\theta_1$ and $\theta_2$ are the lower and upper band-edges, respectively. With the frequency response of the filter

$$H(e^{j\theta})=\sum_{n=-\infty}^{\infty}h(n)e^{-jn\theta}$$

we can write the power spectral density of the output noise as

$$S_y(e^{j\theta})=S_x(e^{j\theta})\left|H(e^{j\theta})\right|^2$$

and its variance is

$$\sigma_y^2=\frac{1}{2\pi}\int_{-\pi}^{\pi}S_y(e^{j\theta})d\theta=
\frac{1}{2\pi}\int_{-\pi}^{\pi}S_x(e^{j\theta})\left|H(e^{j\theta})\right|^2d\theta=
\frac{1}{\pi}\int_{\theta_1}^{\theta_2}S_x(e^{j\theta})\left|H(e^{j\theta})\right|^2d\theta$$

From this equation we can get a lower and upper bound for $\sigma_y^2$ as follows

$$\min_{\theta\in [\theta_1,\theta_2]}\left|H(e^{j\theta})\right|^2
\frac{1}{\pi}\int_{\theta_1}^{\theta_2}S_x(e^{j\theta})d\theta\le\sigma_y^2\le
\max_{\theta\in [\theta_1,\theta_2]}\left|H(e^{j\theta})\right|^2
\frac{1}{\pi}\int_{\theta_1}^{\theta_2}S_x(e^{j\theta})d\theta$$

and from (1) we finally get

$$\min_{\theta\in [\theta_1,\theta_2]}\left|H(e^{j\theta})\right|^2\sigma_x^2\le
\sigma_y^2\le\max_{\theta\in [\theta_1,\theta_2]}\left|H(e^{j\theta})\right|^2
\sigma_x^2\tag{2}$$

These bounds can be useful (i.e. relatively tight) if the noise bandwidth is small compared to the bandwidth of the filter $h(n)$. Unfortunately, they are almost useless if the filter bandwidth is small compared to the bandwidth of the input noise process $x(n)$.

In the latter case, i.e. if the filter bandwidth is much smaller than the noise bandwidth, you can come up with similar bounds as in (2). You just need to estimate the minimum and maximum of the power spectral density $S_x(e^{j\theta})$ within the filter bandwidth.