The first point of call would be to make sure the magneto is actually generating a spark, remove the spark plug, and install it on the lead and rest the metal part of the plug against the engine. Wit hthe throttle in the run position briskly pull the starter rope a few times to see if there is a steady stream of sparks at the end of the spark plug.
If there is spark you would need to look at the fuel system a little more, this can be confirmed by putting a teaspoon of fuel down the sparkplug hole, reinstalling the spark plug and trying to start it again.
In those little Briggs engines you fill it until the oil level is level to the filler hole with the engine sitting flat.
Please turf that filter and put the empty casing back on for the sake of a couple of tests, that foam is ready to break up and it will end up all through the system. a couple of very short runs wont hurt it. The filters are still available relatively cheap, they are not commonly used anymore so I am not surprised that your local shop doesn't keep them in inventory.
