Oh, I thought I had read in your first post that you turned off the port forward to the barracuda.
This is the easiest way to set up the inbound mail:
- disable the NAT
- make the mx record point to the public IP of the Astaro
- add your mail domain(s) to Domains on the Routing tab
- set "Route by" to "Static Host list"
- add an object with the private address of the barracuda to the "Host list"
- for a start, turn "Verify recipients" off
- and, very important: tell your barracuda to accept mail from the internal interface of the ASG
For inbound mail, this is all you need. If this does not work, I would strongly suspect that the configuration problem is on the barracuda (i.e. it does not accept mail relayed through the Astaro).
For outbound mail, you can either send directly, or if you relay through the proxy on your ASG, then you need to add your internal network (or only the IPs allowed to send mail) to the "Host-based relay" on the Relaying tab. But inbound and outbound mail don't actually have much to do with each other.
This is, of course, a very simplistic guide. If it does not get you through, please find the relevant lines in the smtp log and post them here.
Good luck!