Something like nmap might work. That should let you scan your network to find the IP address of your MBWE. Once you get that, you can start connecting to it by IP address.
First find out the IP it took from the router via the router’s built-in DHCP server. Log into the router web interface and find where the leased (given to devices) IPs are listed. Find out which one was given to the My Book and go to that IP with your web browser. From there you can add users and shares, then mount them in Ubuntu like any normal network share.
You might want to change the IP of My Book to a fixed one so that you don’t have to hunt it down every time the IP changes