The ringer probably is a C4A type, and has 4 leads on it. If you can verify that, here are the wiring instructions for such a ringer:
Red lead goes on one of the incoming line wires. Black lead goes on the other incoming line wire. (You can use screw terminals on the network--that block with all the screws--that are convenient, such as L1, L2, G.
The slate (gray) ringer wire and the slate/red ringer wire are attached to the screws marked "A" and "K" on the top of the network. Inside the network, those screws are attached to either end of a capacitor. It allows AC (anywhere from 60 to 105 volts is the FCC standard) to operate the ringer. It also blocks DC that is used for the talking circuit.
There are a multitude of websites that explain all this. Just Google around for "500-type telephone" If you need to see a wiring diagram (The phone company calls them "practices") then Google for "500 telephone BSP" BSP stands for Bell System Practices.
Our friend Paul Fassbender has put together the definitive Western Electric telephone website https://www.paul-f.com
that explains the evolution of all the models, with references to the correct BSP.
Take a look at this one, on page 15.https://www.telephonecollectors.inf...-501-101-i8-ref-500-501-554-556-types-tl