You don't need to know your brother's router's public IP address if you have a DynDNS set up. You only need to know the dynamic host name you created on DynDNS and configured your brother's router to use.
- You have to create a DynDNS account, then a host to use for your brother's Internet connection.
- You have to configure his router to use the DynDNS host entry, by connecting to the account you set up.
- You have to port-forward 4899 on your brother's router to the local IP of the local computer you want to RAdmin to. That's the LAN IP of 192.168.1.x that you mentioned.
Sorry, I can't give you router-specific menu choice, etc. instructions as I don't have those devices.
In your RAdmin viewer on your computer you put the DynDNS host name you created in the IP or hostname field of your connection entry for your brother's computer. When you connect to that you are sending an incoming connection request on port 4899 to your brother's router, which will forward that request to the local computer and you should get a username and password prompt. Enter the username and password you set up on his computer's RAdmin server in the permissions area of settings.
One other thing I strongly recommend is enabling IP address restrictions. It's in the Settings -> Options on your brother's computer. You will need to know your computer's public address. You can use www.whatismyip.com or similar site to get your router's public IP. Enter that in the allowed list on your brother's computer. This keeps the hackers and bots on the Internet from hacking into your brother's computer. If your Internet connection is also a dynamic one (most are) then occasionally when you try to RAdmin to your brother's computer, instead of getting the login window you will get an IP address filter restriction error dialog box. Then you need to go to a web site like the one I mentioned here and get your new IP address, then get your brother to update the allowed list in his RAdmin server settings -> Options to allow this new IP and remove the old one.