I would personally check that capacitor first before even touching the bulb.
That thing is well past the average shelf life for an electrolytic.
That thing is well past the average shelf life for an electrolytic.
So, it might be the 350v capacitor has failed and is not storing enough charge to generate a meaningful flash, or the trigger circuit which is not generating the initial >5kV to get the xenon conducting.
Either way, i'd check the circuit before the xenon tube.