If it passes what grumpy suggested, fires runs and dies out.
Pop the back cover off the carb and check that the cable is rotating the grey plastic thing. Some times they pop out. Check for any cracks in the cover or carb body. Hoses rubbing on them can wear through them.
If it still wont run.
Grab the disc under the mover and see if it has any movement in the bearing.
do this with the plug out. If there is it will need new crank bearings.
If its real leaky it wont start.
Another common problem with not starting is bad compression. A warped head is one cause.
Head down to the local mower shop and get some parts.
Top crank case seal (its a big O-ring under the pull start). Inlet manifold O-rings (big one and little one), take it off the block to fit them. Stop pin rubber. Primer cap (face) O-ring, when you fit this dont roll it on. Put it round the bottom and hold in two spots with your fingers about 1/3 of the circumfrence, then stretch it out to get it over the lip. And if it needs it and new primer face and needle. The brass screw that holds the face on is the main jet, make sure this is clear and not blocked.
Get a some diesel or Kerro and a suitable container. Pull the carb off, strip it down clean everything up.
Blow the body out with compressed air.
Suppose you could just use a can of spray carb clean. Wear glasses that stuff stings like hell if you get it in your eyes. Its bad enough on the face or arms etc.
Put it back together.
Theres a video on Youtube on how to do the throttle cable. Check it out its a big help.
Before you put the carb back on poke the stop pin rubber in and put the pins in with the carb just sitting in the general area. Then check spark. If it passes put the carb back on and it should fire up.
If it was running and now its not i would say the most likely problem will be the O-ring under the pull start. I had one the other day that had ingested the Oring, little bits of it everywhere.
Dont try and use silastic etc on it, just a smear of oil and try and put the starter back on evenly.
Cheers, Bob.
