If I understand correctly, you don't actually want to copy your contacts yo your PC, but rather to back them up (somewhere) so that once you get the new 4S you can upload them to the phone.
As Jakub said, the easiest way to do it is just by using iTunes.
If that is not an option, there are more convoluted ways to achieve your goal by making use of some kind online service, such as a Microsoft Exchange server. That will not backup your contacts to your PC, but rather to a server, which is a good thing because it will ease the transition to your new phone. Instructions for configuring google's exchange server syncing can be found here: http://support.google.com/mobile/bin/answer.py?hl=en&topic=14252&answer=138740
Another option, unfortunately unavailable to you since it requires iOS5 (not available for the iPhone 3G), is to use iCloud as a contact syncing service.