If adding fuel to the cylinder without changing anything else makes it go until that fuel is used up, then the only fault with the engine is that it is not getting fuel from the carburetor. The possibilities are: no fuel in the tank; a fairly large air leak into the inlet pipe; the fuel pump isn't working; or there is a blockage somewhere in the carburetor. You can easily tell whether there is fuel in the tank. If there is, it looks as if you will have to remove the tank and carburetor (they are screwed together) from the engine and work through a process to find the fault(s).

If the bottom of the tank is completely clean, the most likely single cause is that the fuel pump diaphragm has deteriorated, but there are other things that are also reasonably likely. Probably the best solution is to go through the carburetor repair procedure in the B&S manual:
http://www.tpub.com/content/recoveryvehicles/TM-5-4240-501-14P/css/TM-5-4240-501-14P_99.htm
Use the "next" button near the top of the page to access subsequent pages: you'll find there are quite a few pages that you need to read. Please report back here to keep us posted on your progress. Essentially, you probably just need to inspect (and almost inevitably in view of its age, replace) the diaphragm between the fuel tank and carburetor, then clean the carburetor by following the procedure. Be careful to follow the directions when installing the diaphragm.