most all responses have been right on track. I have spent a lot of time shooting rim fire at the range. Nothing competitive but as a hobby. For a "range or precision arm, the bolt action can not be beat. It will be a much better choice. Shooting precision is not done fast where the benefits of semi auto come in. You will need to take your time and use your developed shooting skills. Otherwise you are just blowing off rounds and the results are not going to mean anything. Precision shooting takes a lot more then the firearm. Skills as well as proper equipment heavily comes into play. Proper benches, shooting rests, quality ammo, and proper shooting techniques go a long way. Shooting 200 yards rimfire "is" precision shooting.
Now if you want to just shoot some rounds at 200 (very far for a 22lr) then most anything will suit. But your results will be much better with a proper bolt action
For an inexpensive choice, the Savages are hard to beat. Better trigger (for target shooting) then most in it's class. Decent enough barrel options. Plenty of quality stocks available, etc. For abour $300 or so you can get a decent Savage with a heavy barrel. Now mind you that scope for this is another story as well as all of the other gear. The reason i am mentioning this is I have been there and done that, just trying to save you time and money.
IMHO neither the .22lr, .22wmg or .17hmr are good for 200+ yards. I have used them all. The .17HMR is a tack driver and easy .5moa is realistic at 100 yards. If you get your .22lr to shoot moa then you are typically doing pretty good at 100 yrds. I am not talking once in a blue moon, but consistent groups.
Brands I like or recommend for a non $1000 plus rimfire:
1. Savage with a heavy barrel and easy trigger mods. All of my Savages have had decent 1 to 1.5 moa in .22lr. I have 2 Savages in .17hmr and both drop the moa in half compared to the .22lr's on average.
2. CZ. For a quality gun in the $500 or so rim fire you will not touch CZ. Their bluing, quality of wood, etc are second to none in this range. Triggers are significantly heavier and can have some creep. But with mods, can be inexpensively modded at home. If you want a beautiful rifle that can shoot good, there is no other choice without going much higher in price.
3. Second hand Anshutz. You can find them sometimes around the price of a new CZ. They will be a bit rough at that price. But performance is incredible. They are made with precision as it's goal. Maybe a little strange looking if used to traditional American hunter bolt action look. But if you want to get serious and hit stuff at long distances or achieve great scores, they are a brand to go with.
I would honestly stay away from 10-22's for precision. yes they can be made accurate. but I have seen so many people add so much to them and not get results that match even a stock Savage. I have seen more people at the ranges that are frustrated shooting modded 10-22's then all others combined. not that they are bad by any means. Just that the performance to dollar is high. They are one of the best over all designed rifles ever. But not in the precision role.
Those are just my suggestions after many years of shooting rimfire with many many rifles and brands as well as 10's of thousands of rounds down range. No offence to anyone and any brand they may shoot. I love all firearms...due to the fact they are what they are. But like vehicles, there are some that are suited for certain things more then others. And occasional you get a unique example. I am not going into that. Just going into good imho overall advice on what will make you satisfied at those long rimfire ranges.
If I was going to suggest a street price less then $500 rim fire for shooting distance, this would be an easy choice for a new rifle.
http://www.savagearms.com/firearms/model/MARKIIBTV
I have spent a lot of time on that rifle and it's ergonomics, accuracy and reliability is incredible for the price. I have used that configuration in .17HMR as well as .22lr extensively. Both excellent. I bedded both of the rifles and modded the triggers myself. Sold when I lost my job due to company going out of business. Wish i still had them. After reading this topic I may have to go pick up one tomorrow, lol.