Sounds like you have correctly checked the ignition/spark process.
So it's a fuel issue.
I'd ask myself how old the gas was. If the gas is old - more than a few months - I would consider it too old to use. So I would get the old gas out of the tank, clean the tank if I could, change the fuel filter, change the fuel pump, pull the top off the carburetor so I could clean out the float bowl,put fresh fuel in it and expect it to crank up and idle well as well as reving up smoothly.
The float bowl:
In case you are unsure, the float bowl acts just like the tank on the back of your commode. There is a float, a small valve on the input side and a teeny weeny hole somewhere in the bottom of the bowl for the fuel to get out. Blow out the teeny weeny hole with compressed air - you can buy a can of compressed air at a computer supply department.
If I was doing it, I'd buy myself some gasket material before I started so that I could make another gasket if I tore the original when I took the top off the carburetor.
If you are not too sure about your abilities and don't want to do the carburetor stuff, it'll probably be alright anyway. I just like to go through everything when I get into something like what you have.
Sooner or later it's going to fail. Change it now. Check out the old one and save it for an emergency if it pumps.