This is a bit more responsible of a design,
And yes, for once I'm claiming a 555 is useful for something.
+V can be whatever. For instance, you can use about +100V and use the primary winding inside the FBT itself (trace the PCB to find these pins). For the external winding (usually 3-10 turns), 12-24V will be suitable.
An RCD peak clamp snubber, across the transistor, might also be a good idea, to prevent overvoltage. The RC damper helps, but doesn't limit the peak very much.
I don't have a complete schematic handy, but you would add a diode, resistor and capacitor, connected to the transistor collector/drain just as Q3, D1, C2, RL are connected here:
https://www.seventransistorlabs.com/Images/AltiumDiscreteFlybackSch.pngTypical values would be D1 = UF4007, C2 = 10nF and rated at least 2 x +V, RL = 10k 5W or thereabouts (depending on +V, turns, operating frequency, power level, etc.). You want RL to be small enough that the clamping voltage (the DC voltage on C2) isn't obscene.
Tim