I am doing almost the same thing you are doing. I have my regular iPhone with my monthly plan and then I have my new iPhone that I am using a pay as you go sim card in. Both iPhone have the same data such as contacts and bookmarks. To do this, I just put the old sim in the new phone and I was prompted to restore my phone from the latest backup that I had from my other iPhone. A few minutes later, all my contacts were the same on both phones, same as notes, SMS and all settings.
To do this on your phone, you may have to restore first (don't forget to switch sims) and then choose the last backup that you made on the other phone.