Still have an old but mint steel Weaver 3-9 that I've used on "sporter" Hornets for decdes (it currently resides on my '49 BRNO ZKW465 22H). Good enough for some things, but very lacking for others. But they can still be found for low prices. Also used the "older" Burris 4-12 Compacts. Bought a newer one for the BRNO a ways back and haven't used it, put the old Weaver in it instead. Close as I can come within your price range though, good optics to me starts at a much higher cost, especially if going on a firearm chambered for a cartridge capable of needing higher quality.
My uses for Hornets (22 and 17 Ackley) for decades was mostly longer ranges for from very small varmints to coyotes that demands way more scope than a 3-9 (although in my yourth on the ranch we also killed mule deer with 22H). My minimum starting place is 4-12, with AO, fine hairs and dot being ideal. Have used Weaver KT15's, Weaver V16's, Nikon Monarch UCC Long Range 5.5-16.5's and even a Leupold LRT 6-20 on my Hornets though - all target models with AO, fine hairs and dots. The to hit small you have to aim small consistently thing.
Even if your ranges are not as long as what I consider the 22 and especially 17 Hornets capable of, a resonably priced do all would be the V16. A little bulky on a small sporter maybe, but even my Browning Low Wall 22H wore one quite well (and that's a small rifle). Add another $100 and a Leupold 4.5-14 simply belongs on a sporter and will do most anything a 22 Hornet can. Still have one NIB for the also NIB CZ 452 American 17HM2 I never got around to rigging and shooting before the health failed. Pefect match IMO for the rifle and the abilities of it with that cartridge for squirrel hunting and more.
