Bullet drop on a .22 is similar at 200 yds to .308 at around 500-550 yds. Wind drift is proportionally similar as well. I occasionally do .22 lr shooting out to 250 yds, using 1/4 scale targets of the ones I use for full size distance shooting. It is cheap, fun, and good practice for dialing in proper scope adjustments and wind calls. I've got a .22 lr trainer that mimics my full size .308. If it is something you end up liking, a Savage MkII can be had quite inexpensively and is very accurate (MkII TR model can be had for a bit more).

Ultimately, like most shooting sports, long range marksmanship can be a money pit, but it is a lot of fun. Especially with shorter ranges like 200 yds, precision .22 can be quite challenging and a ton of fun.

With your 5.56, I expect you will shoot soda can sized groups or better at 200 yds with a good rest/bipod and a bit of practice.