How can I back up the phone numbers on my iPhone?
If your whole contacts (Name, Cel, Email) are stored on Gmail, you can use it to sync over the air directly to your phone.

First, create a backup of your contacts and delete the ones you have in your iPhone (this will prevent any merging issues).

Go to Preferences > Mail, Contacts, Calendars > Add Account

Add your Gmail account and when prompted, turn on the "Contacts" option. This will import all your contacts from Gmail directly to your phone.

You're done! your Gmail contacts should be now in sync with your phone, if you use Outlook as your main application to edit your contacts, remember to sync it with Gmail first.