Re: '70 40hp Johnson Runs rough and dies.
I have brought several engine back from the grave yard in my time and have been in your shoes more then twice so I'll try to throw some things out there that I have run into and hopefully it is helpful.
First thing and this is very important, check the compression on each cylinder and report back, throttle should be wide open and choke off when doing this. Each cylinder should be within 10 to 15 psi of each other and if it is under 100 psi you generally have a problem.
Make sure all the jets in the carb especially the high-speed jets are clean! Since you rebuilt the carbs I'll assume this was done properly but it is always worth checking.
Make sure you are getting a spark in all spark plugs, you can have the cleaned baddest engine in the world but without a spark it is worthless.
It sounds like there is a fuel delivery issue if it runs for a bit then dies. A couple things to check...Make sure the gas tank is venting properly, if a vacuum builds up this can cause similar symptoms to what you are experiencing. A quick check is to run the motor with the gas cap off. Second make sure your fuel lines are not sucking air, this can be a very tricky diagnosis and is hard to find at times. Check the o rings on the fuel adapters and if there is rotting or cracking present they are leaking and need to be replaced. It sounds like you have a motor that has been sitting for a while so this is more prevalent in your case. Check all the hose connections and make sure they are tight and snug. I can't tell you how many times I have had a simple problem with air leak or venting issues that is a 2 min fix when I have spent countless hours ripping the engine and carbs apart.
The last thing, and this is more of a QC, is to check all your electrical connections to make sure they are clean, hit them with a wire brush or dremel. Also check to make sure the points are clean and proper clearance, oil or dirt can cause an electrical mishap. But if your getting a spark and the engine is firing this shouldn't be the issue.
As for the fuel bulb being hard through my experience it does not necessarily need to be rock hard. None of mine have ever been rock hard and my engines run fine, its when they get wimpy and feel like you are pumping air when you start to run into problems.
Like I said it sounds like you have a fuel delivery issue. Many people might suggest you adjust the float heights on your carb but I will say in the 10 something years I have been tearing down engines I have never had a problem with float height, it is always another issue that has been over looked. With that being said the previous owner may have adjusted them improperly so they may be at the improper height. This is the only case where I would say to re-adjust the float heights. BUT I would check them to make sure they are within tolerance if you say you have the manual. Unless they are WAY off you shouldn't have a problem with float height.
Another thing to check, assuming you have a two stroke are the reed valves. Take a flash light and see if there is any cracks in the reeds, if so they need to be replaced. This can be rare but it does happen. This can usually be done by just taking the carbs off and using the flashlight but if for some reason you cant get to em you will have to take the intake manifold off which would require a new gasket. When I did mine a few years back the gasket was only around 10 bucks so it wasn't to bad but it can be done without going that far.
Good luck and keep us posted!