Do you have a clear fuel filter? If so check to see if there's fuel in it when it dies. If you don't you could always remove the air filter and squirt a little bit of gas in the carb (very little) and then try starting the engine. If it starts then dies that would also point to the fuel delivery system. Things like the vent in the gas tank could be causing a vacuum or the roll over safety shut off could be stopping fuel delivery.
On these it tends to be either the fuel pump or coil. At 2 hours, it is just getting started and will likely get worse. Good things to start with noted above. As it stalls, is the level of fuel in the filter similar to when it runs normally?
The fuses corrode as they are made of alloy but not steel so they won't 'rust' as such.
2 Hours is a long time to run before a fault. A heat-related electrical fault will manifest itself earlier as the engine is at full operating temperature after 30 mins. So I tend towards the fuel system and agree with the suggestion to check the fuel supply and filter etc. Breather (tank) maybe? Fuel tank suction pick-up can get blocked also...
When turning the ignition key to On/Run you should hear the carburettor solenoid 'click'.
You could check the spark when the fault occurs?
Also, some LT155's had the 'Out-of-seat' relay which is like a delay switch. This can fail.
In addition you should verify the 'Diode' in the loom.