The K20 (RSX Type-S engine) can definately be stuck in a newer civic. It's not exactly easy, but it can be done and people seem to be doing it lately. You just need to get the engine, buy the mounts from
http://www.hasport.com/, and get someone to do the swap (unless you plan on doing it yourself). Just be aware that it'll probably cost you $5,500+ to do this swap, but it'll definately be awesome driving around a K20 civic.
Personally, if I were you, I would just go for a turbo. For the $5k you would need for a K20 swap, you could get a nice turbo kit, build the internals a bit, and be making around 220+ whp, which is way better than the 170 or so whp you get from the RSX engine. Besides, turbo'd cars are much more fun to drive.