Hey ticktock623
You are correct, the PSP’s wireless card is a very old piece of technology that operates on a different language than your home router does. It is normal that the PSP wireless card is showing “Incompatible” when you attempt to connect it to your network. Your router should have whats called a “Legacy Mode”. This allows older devices such as your PSP for instants to connect to the router as would any other device, it just operates at a slower speed and connection type that the PSP can read.
After you change the configuration to allow the router to use “Legacy Mode” connect the PSP to the network. After this is successful run a software update on the system to make sure there is no outstanding updates for the OS and hardware.
Good Luck!