You can use the apps that you have already purchased on the iPad Mini without having to buy them again. You can download the apps again for free as long as you use the same Apple ID that you bought them with.
You can restore from a backup of the iPad and sync with iTunes in order to setup the Mini the same way that the iPad 3 is set up - with the identical content on it. Transfer all purchases into iTunes from the iPad 3, backup up the iPad 3 as well. Then restore the Mini from the backup of the iPad and sync with iTunes and the Mini will be identical to the iPad except that you will have to enter all of your passwords again since passwords cannot be restored to a different device.
How to backup and restore from a backup