[Evan Austin]

Most, if not all of the old, smaller Briggs engines used a diaphragm which acted as sort of a fuel pump, on vertical crankshaft engines it was between the carburetor and gas tank, on horizontal crank engines it was under a plate on the side of the carburetor. Briggs still uses diaphragms on some vertical crank engines which are usually on cheaper mowers. Most newer engines don't even have such a thing. If it has been sitting as long as you say, I would replace it. I would also remove the carb from the tank and spray carb cleaner in every passage I could find, the blow out with compressed air. Also, if your carb is metal, make sure the mating surface to the tank is flat, this area warps very easily.
As another note, on engines that act as yours does, I would try removing the air filter assembly and choking the engine by putting your hand over the air inlet (leave the actual choke off). If the engine dies out, try choking it some more and see if it comes back.
By the way, to see how old your engine is, look at the serial #. On a Briggs, the first 2 numbers are the last 2 numbers of the year of manufacture, the next 2 numbers are the month, the next 2 are the day, the last 2 are the production line the engine was built on. On a Tecumseh, the first number is the last number of the year, you don't get to know the decade. On a Kohler, you take the first 2 numbers of the serial # and add 70 to get the year.