You should be able to do it. I had to play around with the app on my iPhone a little figure out how to load books onto my iPhone. Try looking for something like "archived items" and then you should be able to see the books that you have in your Kindle account. Then you can select which ones you want downloaded to your phone.

I tend to have different books on my phone than on my Kindle, so can't comment on the syncing part of it.