With automatic sync on, your changes go to your iPhone about every 6 hours. New iPhone contacts don’t come in on their own. To bring them in, tap ↓ Get new on your iCloud row in Sync / Import.
Fix a person once on their Nimbus card — it shows up on your iPhone at the next sync.
The app-specific password never expires on its own. If you change your Apple ID password, Apple revokes it — make a new one and Connect again.