Newb guy

Ok all you KZ 650 guys. I just picked up a 1977 kz 650 (B1 I think). The PO did a lot of work bringing this bike back to life. I test rode it and it ran good. Then I got home. It ran for about a minute then acted as if it was out of gas. One thing I didn't look at when I bought it was the inside of the tank. What a unholy, rusty mess. I have no doubt that some of that crap not only clogged the petcock but likely also made it's way into the carb. I've never seen one of these carbs...ever. I want to rejet it after cleaning because he said it has the stock jetting and it came with pods. No air box. I see the main jet. Then to the side is another jet. Is this the pilot? Everyone says go up a size on the pilot. (Of course a bigger main jet as well. It's the carb with the needle jet thingy on the bottom that you can adjust front the bottom side of the float bowl. Any help explaining the basic process of replacing the jets would be helpful. I've rebuilt a couple newer style carbs but I lack the wisdom of the ancients that designed these perplexing devices. Thanks all! (Visual aids greatly appreciated)

My suggestion would be to clean / derust the tank first.  Replace or rebuild the petcock at same time.  Then see how it runs.  
Then go after the jetting / carb issues in phase 2.