What is the input to the recorder? Is it high impedance like a 3.5mm phone jack or low impedance like an XLR connector? Balanced or unbalanced? What is the required level, i.e. "line level" which is usually hundreds of millivolts or "microphone level" which can be a few millivolts and sometimes less.
A lot of small condenser mics have a built in FET amplifier (really an impedance converter) and give you a few millivolts (from 90 dBA) without additional amplification.
If you want to ignore anything above 1300 Hz, you're going to have to explicitly filter the signal, probably with an op-amp active filter. The good news is that you can design any required overall gain into this same circuit. Something like an LM358 or LM324 plus some resistors and capacitors. You'll need more information than you have shown here to design a filter. Simple electronic filters don't have a brick-wall cut off frequency. How much attenuation do you need at and above 1300 Hz? How much attenuation can you tolerate below 1300 Hz? It sounds like there must be some very specific sound you need to capture. What is the actual frequency band of interest? What is it that's above 1300 Hz that you need to avoid?
Forget about the distances. What you need to know is the sound sound level (usually in dBA or dBC) or the sound pressure level (usually in Pascal) at the microphone.
Three meters from the amplifier to the recorder is no big deal. High-quality, low-impedance mic cables can be 100 meters long.