Could have been a Model 510 Targetmaster Bolt Action Single Shot or any of others. The Remington 510P version was my first rifle given to me in 1958 used and I still have that rifle. Yes, it is among my most cherished guns.I don't know the model number offhand, but it's a Remington single shot bolt action .22 that shoots CB Caps, BB caps, Short, Long and Long rifle cartridges. Many in the day required manual cocking after the bolt was closed, this one cocks as the bolt is closed.
My mom bought it for me in 1959. She knew a jeweler who was a shooter and she got me into a training class. That started it all for me.