As I understand matched filter is optimal linear filter for maximizing the Signal to noise ratio (SNR) in presence of noise.
some restrictions on the noise apply (mainly: should be additive), but yes!
Consider a radar/sonar system with narrow-band pulse
A pulse that is narrow band is not really much of a pulse, if you'd ask me, but that's more of an opinion but a scientific fact.
However: the smaller the bandwidth, the longer the duration of the pulse. Basically, for pulse radar, bandwidth $b$ is inversely proportional to pulse duration $T$.
specifically for range resolution
low bandwidth is very bad for range resolution. A bit of fiddling with the radar equations will show that your resolution is bounded by the half distance your wave travels during $\frac 1b$ (monostatic rada). Now, as mentioned above, for the pulse radar, that leads to a resolution of $c\frac 1{2T}$.
As per my understanding, under this condition we cannot resolve if there are multiple target within the pulse, my first question: is it correct?
Exactly. See statement above.
However! You can get something that is equivalent to a higher bandwidth by multiple observations under coherent conditions. I'd doubt you'll easily distinguish two targets by that without a secondary source of information, though.
We can detect multiple target within the pulse if we use linear chirp with reasonable bandwidth
That's why people use chirps: The same bandwidth $b$ as a pulse of duration $\frac1b$, but you can spread the energy over a larger time $T\gg \frac 1b$. So, chirps, unlike radar pulses have sub-1/signal duration resolution, because their duration is not inherently linked to the bandwidth.
Is anyone in the community is interested to resolve close target with narrow-band pulse in the expense of heavy computational cost?
So again, you do not want to go narrowband and pulse at the same time. That's just the worst of two worlds.
What you do is apply proper signal processing gain and knowledge on the target as the signal to get higher resolution than a single transmission of given bandwidth would have, and mostly, you combine a lot of observations.
For the "far far away" target, I'd encourage you to learn what a synthetic aperture radar does, in terms of signal-processing. There's pulse compression (like you know it from the chirp) and there's the combination of a lot of observations to mimic a really narrow beamwidth and get a resolution that is far higher than one could expect from a single observation.
For the closer target: OFDM Radar is your topic. You can use the existing OFDM waveforms, for example of vehicular IEEE 802.11p "WiFi", and exploit the fact that you know a lot about these and have very good estimators for channel state that you can apply on the reflections. Martin Braun wrote a dissertation on that (which, by the way, also does a very decent job of explaining typical boundaries in estimator's variance, accuracy and radar estimation possibilities in general).
