A used RWS 34, HW 30 or HW 50 would work. A used HW may get you in your price range. I like the RWS 34 as most bang for the buck,. Good accuracy, good quality and power enough for small game hunting or pest control.
If you’re in search of an effective rifle to take down small game at a range of 70 yards, the Diana RWS 34 .22 caliber airgun is an excellent choice.
thegunzone.com
https://www.airgunsofarizona.com/spring-piston/rws-model-34/ -$255
Flying Dragons use to have a RWS 34 Chinese clone from Xisco that was cheaper and good quality. Not sure if they still do as previous poster said website is down for reconstruction. Suppose you could call him. I would pay to get it supertuned by Flying Dragons which ups the price. Highest UTG scope you can afford as far as scoping any airgun. I like the UTG 3-12 Bugbuster.
As far as a pumper, I would get or borrow a cheap Crossman 760 or better yet the Daisy 880 from Walmart and see if he likes it before going for a quality pumper. The pumping does get old but is fun for me as it reminds me of my childhood as I have the same exact guns I had or wanted as a child. If I was just starting out in airguns, I would have gone solely with springers.