There is likely a bare wire from the neutral bus going to a water pipe (and possibly a ground rod) somewhere, and possibly to the kitchen and bathroom outlets. The neutral bar in the bulldog panel serves as a ground bus also.
My 1960 house in Santa Clara had a bare #8 copper wire running under the stucco to the nearest water spigot -- very difficult to find.
When you are done, there should, at a minimum, be un-spliced ground wires from the main panel to both the underground water service pipe and a ground rod, or two ground rods if the water pipe is not metal.
You'll need both a ground and neutral wire running between the main panel and sub-panel. In the sub-panel, the neutral bus must not be bonded to the panel and ground bus. Have you considered upgrading your panel? The bulldog/pushmatic stuff is dated and considered a fire hazard by many (mine is long gone!!!).
What I mentioned here is not everything you need to know to do what you are talking about -- just the beginning. Please make sure you take the time to really learn this stuff well before doing any work. Do a lot of reading, ask for help, and be careful. I would strongly consider having an experienced electrician (not a handyman) do/supervise the work. You are getting a permit and inspections, right?