This is the ballast schematic of this lantern.
As you can see, it is rather basic blocking oscillator inverter.
The secondary is polarized so, during the flyback the side with the 2V filament winding is negative.
When there is no arc in the tube, the inverter generates rather high voltage flyback pulses. Those pulses are present across the filament winding and as they shape means quite high (for the "2V" winding) rms voltage, so the filament get heated. At the same time the cold, so low resistance filament presents quite significant load for the circuit, reducing the peak voltage.
When the filament heats up, the cathode starts to emit electrons and as the filament resistance increases, the peak voltage across the lamp increases, until it ignites. As only one side is heated, the lamp then acts as a rectifier, letting just the flyback polarity pulses to pass. That is important to decouple the battery voltage from the arc voltage: When the transistor is ON, so the voltages are derived from the battery, the tube does not conduct. When the transistor is OFF, the tube is polarized with the heated electrode negative, so the arc conduct the current. But then the battery is disconnected, so the tube is free to dictate the voltages.
The C2 is there to limit the peak voltage during the ignition attempt, as well as to recirculate the energy back into the transformer, when the lamp does not strike yet, so reduces the power dissipation and battery current during that mode, easing the startup.
the trimpot is there to factory-adjust the desired power, as the transistor gain tends to vary a lot among production and this circuit is quite sensitive to that. It adjust mainly the OFF time, together with the C3 and winding ratio.
The C1 is there to limit the reverse emitter base voltage, as transistors usually do not stand more than 5V.
The R1 limits the base current, when the flyback pulse drives the base into the voltage breakdown (during an ignition attempt).
The components values are based on their marking.
The transformer "voltages" are intended to show the ratios (correspond to voltages, when the transistor is ON with 4.9V battery).