Congrats on the rifle. I just bought the 512 carbine version. Hopefully will be shooting it for the first time saturday.
I have many bolt action cz's from 30-06, 6.5x55, 7.62, 17hmr and many in 22lr.
I used clp to clean them. And sometimes hoppers as well. I don't use a rod. I use the worm patch made by 2020 concepts. It's easy and works. I don't like the hopes patches I use the cotton ones which I think absorb better. I took my 512 apart and to my surprise there was not a lot of grease in the inside just light oil. So I left the oil and cleaned the bore. This was also prettty clean. Hope this helps a little.