If you select your iPod in iTunes and click the Music tab, you can choose which playlists you want to copy to the iPod. But the other posters are correct — copying a playlist will not cause the songs to be re-copied to the iPod, unless the songs have changed since the last time they were copied (new album art, say). And if that happens, the old ones'll just be overwritten on the iPod.
My iPod contains all of my playlists, but only one copy of each track. I.e., the music library on my computer is almost exactly the same size as the one on my iPod. If each playlist had a unique copy of each song, it wouldn't all fit on my iPod!
Also: if all of your music is already on your iPod, it shouldn't take more than a couple of minutes to sync the iPod, in which case all of your playlists will show up just fine. The only reason it would copy the songs again is if your iTunes tracks are different from your iPod tracks — new artwork, newly updated for gapless playback, edited fields, etc. If that's the case, letting it sync for a long time once will then enable you to sync quickly in the future.