Port 6346 test will show that your router & ISP is not blocking that port for outgoing requests. But for incoming, the airport still might be at fault.

Who is your ISP (internet service provider)?
Yes there is a chance they're using a special anti-p2p program to detect & filter out p2p programs to either affect their bandwidth or stop them connecting.

BTW I've never used an Airport myself & don't know the differences between models, etc. But did you check your airport manual for how to access/create/change your admin password for the device?