Yes, that is the whole point of iCloud. Starting with iOS 5, whenever you power up a new phone, it asks you if you want to set it up as new phone or restore a previously backed up phone. The backup remembers your phone settings, call/text history and a list of apps installed. Once the settings are re-applied, the phone will start re-downloading all the apps that were installed on your backed up phone. You may need to manually copy all your music. Although if you have iTunes Match, it will automatically pop up on your new phone.