How can I test a ballast circuit without?
Different ballasts have different strategies for delivering the initial high voltage, then a reasonably constant current.  It will be difficult to test the ballast without knowing its individual specs, and difficult to repair if broken.

It will be easier to test the lamp in some cases.  If the lamp has two pins at each end, then connect a ohmmeter between the pins, separately at each end.  There should be a heater, like a lightbulb fillament, between the pins.  For a small lamp like yours, the resistance will probably be in the 10-100 k&Omega; range.  No matter what, it should be clearly distinguishable from a open circuit.  If the ohmmeter reads open circuit, then the heater fillament broke and the lamp is no good.

If the lamp has a single pin at each end, then you can't reasonably test it yourself.

Otherwise, just get a new lamp and see if it works.  They aren't that expensive.