You can't humanely kill anything with airsoft, they are made to only put out 1 joule (0.75 Footpunds). It will be very hard to kill anything with this from range.
If you are just looking for quick and cheap pest control then a springer is the best option. Easy to use, hand cranked so no extra equipment needed and cheap.
You only need 2 things with springers, ammo and the gun, if you want to get fancy a cheap scope rated for springers can be had for a few bucks.
As for noise you'll have to attach a suppressor if you think the normal blast is too loud, you'll not need any hearing protection regardless.
For about $200 you should be able to get a decent gun, ammo and scope. You can't just put any scope on a springer though, their recoil does more damage to scopes than .50 BMG due to how they work, seller will know what works if you go to a store, 4x scope should be less than $10 (the one I have cost roughly $4).
You can get a zero recoil springer like the Norica ZRS (Zero Recoil System) but the rifle costs 2x that of a normal springer. A spring suspension scope mount would allow you to add any scope you want.
As for caliber 4.5mm (.177) is good enough for mice and rats up to 25 yards.
If you want a PCP then I would suggest the Air Venturi Avenger, it's an excellent gun for he money. You don't need anything above $500 for mice and rats.
The problem with PCP is the extra equipment you need, you can't use a normal compressor to fill it but a special purpose built one. There are cheap chinesium compressors for around $200 that will do the job or you can get a hand pump for around $100.
So the question comes down to how much you want to invest. With a springer you can get everything you need for $200 or lower and they are easy to care for and very reliable. As I kid I had a Gamo springer that I probably shot over 20 000 shots with over 10 years. I didn't properly take care of it, it was starting to rust in the end but it held it's accuracy at 10yards for over 10 years without me ever sighting in the scope after the seller sighted it in for my.
PCP are more of a comfort thing, they are really good if you shoot a lot. Unless you have a mice/rat infestation I wouldn't recommend it, then again mouse traps could probably solve your problem regardless. PCP rifles are more expensive, require a lot more maintenance and equipment but are more comfortable to shoot once everything is set up.
TLDR:
Spingers: Cheap, reliable and basically zero maintenance. Enough for 10-25 yards.
PCP: More expensive and require additional equipment to charge the rifle. Once you have everything they are much more comfortable to shoot than springers.
What you need to ask yourself is what do you actually need? Would mousetraps be the best option? If you don't know anything about airguns and have no real interest outside get a pellet gun for pest control then I'd go with the springer. It's a "cheap" investment and if you want to delve deeper into airguns you can get a PCP rifle later on.
Springer = Spring loaded rifle
Gas piston = Same as springer but uses a gas piston instead of a spring
Pump = Basically a PCP that you handpump for each shot, depending on rifle it might be 5-10 pumps per shot - I'd never recommend these to anyone unless there has been some sort of mechanical advancement that has revolutionized these compared to the ones I had as a kid 30 years ago.
Co² = Airgun that uses Co² cartridges to power the airgun. Depending on rifle you can get 20-50 shots per cartridge
PCP = Pre Charge Pneumatic, a rifle that has a high pressure air tank as it's power source. Only annoying part with these will be how to fill the air tank, compressor or hand pump. You will need to change o-rings as they wear out but that's probably once every 1-2 years depending on wear.
My recommendation would be to get a springer because of how easy they are to use. If you don't know much about airguns it's where I would start.
You could also get a Co² powered bb gun but not sure how good they are for pest control, then again you can get a full auto one with 50 bb magazines but spray and pray isn't exactly humane.
