Can an automatic gearbox improve engine transmission for an ultralight?
A gearbox is not a useful addition. Some aircraft have geared propellers, but that's a single, fixed gear. 
The engine has to be able to turn the propeller when it's in the highest gear you'll use, so that determines how much power is required. 

In a car or motorcycle, the lower gears are used to match engine speed to wheel speed, allowing the vehicle to drive at lower speeds than 'idle rpm in top gear' without slipping the clutch or the wheels. 

In an aircraft, that function is not needed: a propeller can slip freely.