The device is a phone network controller. It has in internal capacitor that according to classicrotaryphones.com is a 0.47µF capacitor.
The ringer must have a capacitor (condenser) in series before connecting to the line. The network provides this 0.47µF capacitor on terminals A and K. Polarity is not important.
I am just unsure of a smaller equivalent version/option.
Interestingly, the symbolic in the drawing indicates a electrolytic capacitor....
Usually will not harm if you replace an electrolytic capacitor with a non-polarized one.
But the bipolar one will be bigger and may not fit in the block thing
If you just want to make the ringer ring without a connection to a phone line then you need about 75VAC around 25Hz, exact voltage and frequency not important. The capacitor is to isolate the 50VDC on the line from the bell. No line no need for a capacitor.
But no, your description does not make sense, I'm not even going to try to understand what it means.
If you just want to ring the bell without it being connected to a phone line or the other functionality of the phone then you need to generate about 75VAC at 25Hz and connect that to the bell. I can't imagine where you think your capacitor and wires from the Arduino go.