What options are there for permanently connecting a rooted Android Phone to a NAS?
First, you need a "DynDNS" account, so from outside, you can connect to your home network without knowing the (daily changing) public ip.
Then, you install ssh on the NAS (I guess you will use linux, e.g. ubuntu server lts).
I'd recommend forwarding another port to ssh, like 12345, in order to stop at least basic, automated scans from finding your ssh server.  

As client, I would recommend ES File Explorer.
Connect to your NAS by sftp.  It's using strong encryption.