Whether or not you can jailbreak your new phone depends on the firmware it comes with out of the box. If it is on 6.0-6.1.2, then yes, you can jailbreak it using the latest version of evasi0n.
To back up your info, plug your phone into your computer, open iTunes, then click the "backup iPhone" button. This will NOT back up any of your cydia data. To do that you need a tweak such as XBackup (paid) or PkgBackup (free) from Cydia, but this route is not recommended. Mass-installing tweaks can lead to dependency and conflict issues.
I have found the easiest way is to take a screenshot of your installed tweaks list (home+power for 1 sec) and the source list (only really needed if you were using beta repos). Then when you get your new phone, jailbreak it, then plug it in and let iTunes restore from backup. The screenshots you've taken will be on the camera roll. You can use those as a reference to restore your cydia tweaks.