You can configure many routers to act as DNS servers, then advertise themselves to client devices as a DNS server (with a secondary like 8.8.8.8 or whatever). Then you can just use the hostname of the device you want to access and as long as your client device is using your router for DNS it will resolve correctly.
How specifically to set this up depends on your networking equipment
I can’t compare to fastmail since I haven’t used it, but I’ve been very happy with proton mail.