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.
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.