The likely failure modes here, assuming that you have wound the transformer with the correct phase relationships for each winding and also wired the correct ends of each winding to your driver circuit, is either excessive voltage on the base of your driver transistor, excessive tube current or lack of resonance. A better transformer will help, but it is a poor circuit to begin with. One major issue with this circuit is load dependent resonance, this will vary lamp to lamp to a large degree and will affect everything from initially striking the lamp to keeping it lit as well as the current drawn. Dimension the components and transformer correctly for the lamp being driven and the circuit will resonate, the current drawn will drop after the lamp strikes up and the transistor and transformer will run reasonably cool. Get it wrong and the circuit will not resonate properly, if at all, excessive current will flow in the primary, leading to heating of both the transistor and the transformer winding and the lamp will either not strike at all, or may initially strike then run really dim as the circuit struggles to achieve resonance. Another thing to be aware of here too, is generation of stray radio waves, this circuit is likely to cause major radio interference locally.
My advice here is to look for a better circuit, perhaps something using a more traditional push/ pull transistor arrangement and a much more forgiving transformer design
You haven't stated in your profile where you are in the world, but there are inexpensive drivers from the far east available on ebay, amongst other places, that come fully assembled and are reasonably reliable, maybe that would be a better route for your project? All you have to do is add your tube and a power source