Yes, for the NFC tags you will need an internet connection, but the Neucards app does maintain a local cache of the digital cards you have received. You can view the details of those cards and kick off an action, like make a call or send an email, by just tapping on the info. Updates occur when I edit my card or account information and notifications are sent to the list of recipients for my card. Neucards does a sync at launch or when triggered by a notification.

As for adding info to the phone contacts, this is a tricky situation. Right now, I do not sync the contact info from received cards into the contacts on the phone. This is because the contacts on the phone are a shared resource that other apps can access, and I don't want people's contact info to leak. I do support caller identification, so you will see a person's name rather than just a number when they call for all of the numbers on your received cards. But, this isn't as robust of an experience as having all the info in the phone's contacts.