Not all that familiar with Puppy but it has a few utilities under networking to find public shares on the network, pnethood, yassm samba share search and yassm share mounter. pnethood found my NAS but not the browseable pulic shares.
yassm share search found and you could use the share mounter but I did not play with them. Enter the credentials for username, password, share name etc.
Manually mounting worked too from the command line.
Code:
mount -t cifs //router_IP_address/share_name /mnt/point -o vers=1.0,sec=ntlm,username=your_name
You will prompted for your NAS username and password or if none use the guest option. I am guessing your router is still using CIFS version 1.0.