Finding Failed Component in broken florescent lantern circuit
I agree in that the transistor is most likely the damaged component.  One simple way to verify that is to replace it.  If the lamp works, then it was the transistor.

You don't say what kind of transistor it is.  Different types are tested differently with ohmmeters.  For example, to test a NPN first find the base.  This will be the one terminal that looks like a diode to the other two.  For a power transistor, the collector is the one that will be connected to the case or the heat sink tab.  The other is the emitter.  For a small signal transistor, just pick E and C arbitrarily for now.  Now have the ohmmeter apply a voltage accross C-E.  The resistance should read inifinite or very high.  Dampen two fingers and put them between C and B.  This should make the transistor turn on partially, which should show in a obvious way on the ohmmeter.  To verify that amplification is going on, connect the same damp fingers between C and E and see that you get a much smaller response on the ohmmeter.  If you're not sure about which is C and E, filp them around and try this test again.  You will get somewhat higher gain when C and E are hooked up correctly.