It's called Castle Rock, and it's one of numerous examples of basalt rock around the area that are from lava flows millions of years old.
It's also sometimes called Queen Vic by locals, referring to Queen Victoria.
There are a couple main accesses to Arch Cape and a handful of smaller, hidden ones. Both involve parking in neighborhood streets next to or in front of homes, so be cautious and courteous. The best access is the southernmost one, right next to the Arch Cape Tunnel. This one has the most parking space and allows you to be unobtrusive to locals.
Below: looking towards the Arch Cape Tunnel on a particularly misty late summer dusk.