You should consider looking into a lever action. Even though it isn't a single shot it is probably the closest thing to ambidextrous as you can get. A break open is a single shot and is also pretty easy to use with either hand, but I dont know if anyone still makes one in .22. I know that marlin makes a .22 in lever action. I bought my first .22 from them when I was around 10 which was 8 years ago, and it was only $100 new for a semi, and I haven't had any problems with it yet, except that its getting a little small.

http://www.marlinfirearms.com/