Hey Xihiroth,
You should be able to do this I'm sure. If I'm wrong someone please correct me.
Before you hand your broken iphone back in, makes sure all of the apps are transferred to your itunes library. You can do this by right clicking on the iphone in itunes, and clicking Transfer Purchases.
Also right click again and click on Backup.
Another thing you want to do is make sure all of your contacts are backed up as well. You can easily do this by syncing to Windows Address Book if you are using Windows.
Once you have backed up all of these you should be able to plug your new iphone in and set it up.
I 'think' you can restore from the latest Backup that you took, which will put your settings and stuff back on the phone, and then Sync to put the contacts and apps back on as well.
I just read that your phone is JB'd... I recommend Pkg Backup in Cydia. It backs up all of your Cydia downloaded apps, either to a single Contact, or to Dropbox if you use it. So if you end up using this, run a Backup as well in Pkg Backup, and if you back up to a Contact, make sure you do it before the previous things I mentioned so that the backup / contact list is saved properly.
Good luck!