If it's shutting down quickly after priming then it's not getting fuel or enough fuel. Leaking crank case seals can cause this because there isn't enough vacuum and positive pressure to make the diaphragm push enough fuel. The stiffer the diaphragm is the more pressure it takes to pulse it. This condition can also cause a poor lean running condition. There is a procedure for doing a leak down test near the end of the engine rebuild thread posted by webhead.
Leaking carb gaskets could also be an issue. Spray a little carb cleaner on them while it is running, if they are sealed you should hear no change in the rpm. If rpm does change, they're leaking.
If you are twisting that carb adjusting needle open quite a few turns and it isn't changing or flooding the engine, it's not getting enough fuel so it could be clogged somewhere or the previous stated conditions.