You can check the fuel pump, disconnect the hose that goes to the engine crankcase. If its a twin cylinder, it connects to a valve cover, if its a single cylinder, it usually connects to the dipstick tube. put gas in the tank, If gas comes out the nipple on the pump, or the hose from the pump, you need a fuel pump.
It is very rarely the fuel pump.
You can also disconnect the carburetor from the manifold, hold or support it in its proper orientation like it was still on the engine. Put fuel in the tank. If fuel drips out the throat of the carburetor, the problem is the needle and seat (or the o-ring above the seat, if its a Nikki carburetor.) In some of the carburetors, the seat is not replaceable and if a needle does not fix it, you will need a new carburetor.
In the lawnmower shop, I usually specify a new carburetor, it is much more likely not to come back on me if I just replace it.