Firstly you need to realize what information the question has given you, namely that the sphere radiates 4.3 J/s^2. Recognize that this is a power (Power=Energy per unit time) and you have an expression which relates some variables with the Power it will emit.
Since it does not give you explicitly any information on the parameters in P=σAeT^4
but it wants to know what happens if you area doubles, you will need to use some algebra.
So if the right hand side of that equation doubles, then so also does the left hand side. You need to work out what happens when the given values change.
Really what you are getting at here is how things scale, so for instance given the equation velocity=distance/time I know that if I double the distance I have to drive, then to do it in the same time I will need to go twice as fast! This is a linear relationship, or directly proportional between the distance and velocity (if we constrain ourselves to making the journey in the same time)
A more complex example is the area of a square say, where A=x^2 . Then if I double x, the area will increase by 2^2=4 so this relation is nonlinear.
Hope that helps