The most common cause of engine surging (rev up and down) is a dirty air filter.
Lean Misfire
My engine is idling too high. How do I adjust the idle speed?
check you tube under removing the govenor on a Briggs and stratton
Top no-load speed is 3050 rpm.
All Briggs engines run at 3000 rpm. Slacking off the 10mm nut on the gov lever adjusted the arm of the gov to which ever speed you whant but be careful not to over do it
On a horizontal crankshaft engine it is done by loosening the pinch bolt and rotating the shaft in a clockwise direction with the throttle in the high speed position and retightening
4.5 mower will start and run for two seconds
Depending on the model, 3600 - 4000 rpm top no load speed.
Clean the carburetor with special attention to the high speed jet and fuel passage.
mower carbs built since around the mid nineties are non adjustable, except for idle speed,
it depends if you want speed or power, what cog and chain you have and how big they are (small on motor and big on axel gives you lots or acceleration and power where big on motor and small on axel will give you a very high top speed (if you can get it to move))
Allow the engine cool off for 15 - 20 seconds. Position throttle to full speed and turn ignition switch off. This will reduce the vaccum through the carb thus drawing less fuel into cylinders.