How to swap contacts to my new iphone without using itunes?

I just want to get the contacts from my dodgy iphone4 onto a new iphone4 but when i back up the old one and then plug in the new one and click on the manual restore back up it doesn't show that i have the option to use the back up i've just done. Is there any other way of swapping my contacts over?

You can't

Yes there's, download an app called My Contacts Backup on the old phone with the contacts that you want to transfer over. Then in the app tap backup then tap email and email it to yourself. Now on the new phone go to your email that you sent it to and open the attachment and it should transfer over.

Use icloud go to setttings>icloud>then enable contacts. After you put it in icloud when setting up your new iphone 4 enable icloud, or if you all ready set it up backup your iphone

Syncing your iPhone
1. Connect your iPhone to your computer.iTunes opens automatically. Locate and click your iPhone in iTunes under "Devices," then click the "Info" tab.
2. Click the tabs associated with "Contacts" and other data you want to sync every time you connect your iPhone to your computer.
3. Click "Apply." Your computer syncs your contacts and other information you've specified with your iPhone. Manually Restore Your iPhone
4. Connect your iPhone to your computer.iTunes opens automatically.
5. Control-click or right-click the iPhone icon under "Devices" in iTunes.
6. Choose the "Restore from Backup" option. This option will restore your contacts to your iPhone. Your iPhone and iTunes will now be synced, without the loss of data if you deleted contacts from your iPhone.