Yes. Sign in to the same account in the app and at castria.app, and these follow you everywhere:
- the shows you subscribe to;
- where you are in each episode;
- your Up Next queue;
- which episodes you've played;
- saved episodes and playlists.
Downloads don't sync. Each device keeps its own downloads and its own download limits. With Castria Premium, saved episodes are also kept in your cloud storage, so you can download them again on any device.
If something looks out of date, close and reopen the app or reload the page, and check you're signed in to the same account on both.