Do I need a siphon to empty the fuel?
People here are posting advice about everything except what you should do to get your mower running now.
If the gas in the tank is old, you should empty it. The simplest way to do that is take the gas cap off and tip the whole mower on its side. Or detach the tank, to do that you'd likely have to remove the plastic trim cover on the top of the engine. You could of course also use a siphon.
Then loosen the bolt on the bottom of the carburetor to drain any old gas out of there. With fresh clean gas in the tank, confirm that fresh clean gas drains freely from the carb.
My no-start troubleshooting process starts by removing the air filter and putting a 1 second spray of starting fluid or about a teaspoon of fresh clean gas into the intake. Then pull the starter up to about 5 times. One of three things will happen:
A. It starts and runs. Hooray! You may be OK for the whole season now. If you have th repeat the priming process on the next cold start though, the auto choke is not working as others mentioned.
B. It starts for a couple of seconds then dies. This means that fuel is not flowing through the carburetor. Remove the carb, disassemble it and clean the interior with spray carb/choke cleaner. Pay particular attention to the main jet passage that goes up through the center of the unit.
C. It still does not fire at all. The fuel system is likely OK but there is a problem with spark or compression (the other two elements of the holy trinity of internal combustion). Or the engine could be badly flooded.. Troubleshooting this case would continue by removing the spark plug.