For powering something like that with dodgy vintage wiring and insulation, you really want an isolating transformer, not an autotransformer. Just pick up a cheap site transformer and use it for any other 110-120V kit you acquire. Its not like you will be putting this gadget into regular use, so why tie up funds for a dedicated supply?
LOL at being safety conscious with a device that emits electromagnetic radiation from UV all the way up, potentially, to soft x-rays...
It would only generate X rays if applied to a device containing a vacuum.
Last time I have seen one of those devices being used was in a high school chemistry lab for sparking chemical reactions.
Perhaps the simplest solution would be a series cap or capacitive voltage divider? Since the device is inductive, beware that the required capacitance would be lower than what the current would suggest.
It won't generate any x-ray, to generate x-ray it requires
(1) high voltage DC (30kV - 140kV) used to accelerate the electron
(2) vacuum tube
(3) a suitable anode target material such as Tungsten or Molybdenum Target
(4) a filament (thermal electron emission)
The x-ray photon is generated by bremsstrahlung effect
(1) X-rays can be produced by voltages as low as 100 V, but 1 mm glass will block most of the x-rays below 10 keV.
(2) X-rays can be produced in a tube with gas, as long as the gas pressure is low enough that electrons can accelerate to a high enough energy before hitting anything. This pressure is much lower than typically found in violet ray tubes though.
(3) Most materials will emit bremsstrahlung (and characteristic x-rays) when hit by an electron, molybdenum and tungsten are typically used because they have a high melting point (allowing a high power density and therefore a small focal spot for a given input power) and a high atomic number (giving high bremsstrahlung production efficiency).
(4) Most common x-ray tubes today are based on thermionic emission, but tubes based on gas breakdown and field emission also exist. Gas filled x-ray tubes were the only type in existence between 1895 and 1912, when the hot cathode tube was invented by Coolidge.