Derive Parameters for Unknown Flyback Transformer
To determine the core saturation and permeability you probably need a function generator with the ability to generate a high enough voltage/power. 

I am not sure what the primary looks like in your case but I am thinking  of an audio amp or maybe the type of amplifiers EMC people use in there test. The latter are expensive and not easy to come by for the average hobbyist. 

Use an oscilloscope to measure the current over a series resistance to determine the impedance and saturation point at different frequencies. Then work you way back through the typical solenoid formulas do determine the core properties.