Transfer contacts from an iPhone 3GS to an iPhone 3G

No, "backup" is referring to the OS, Apps and phone's settings. Contacts is a separate entity.

Going through iCloud is probably the eziest and less disruptive to both of you.

If she already has her contacts in the iCloud, this is extremely ez. Just configure working phone iCloud and sign in with her account, tell it to sync iCloud's contact. It will ask whether you want to erase contacts already in current phone, u go YES. Done.

But I assume she hasn't used iCloud, then you have to download her shattered phone's contact into your iTunes, THEN, upload to the working phone. This may mess up what you already have in your iTunes. That's the gist, I don't do this everyday to be able to give u a step-by-step.

The iPhone 3 is not compatible with iCloud, the highest iOS version is 4.2.1, iCloud was introduced in iOS 5.

OP, which is your sister's carrier? If AT&T, you can take both iPhones into an AT&T store and explain to them what you're trying to do. They can take the contacts off of her old SIM card and transfer them to the other functional iPhone SIM. It's the easiest way to deal with two older devices given your circumstances. Other carriers may be able to handle likewise. They can also set up the functional iPhone so that it's working before you leave the store.

You can use itunes to copy and transfer the contacts from the broken iphone to her computers contact list.
Plug it to itunes and select the iphone on the top right and then go to the info tab.
Then select Sync contacts with:
And then hit sync to transfer them to your computer and from there plug in the iphone 3G and sync them back to that one.