Schematic looks like it has some crossovers, etc. The path for the neon bulb is: 160 VDC --> Neon bulb ---> Transistor 'C' ---> Transistor 'E' ---> Emitter resistor ---> GND. Then connect the transistor base to your Arduino device.
If you are driving the LEDs directly from the Arduino, be careful with the current. I dont know how much current your LEDs require vs the output current of the pin.
For example, if you have 3 LEDs that run at 10mA each, that's a 30mA load for the Arduino, which is pretty high.
As an alternative, you could connect all 3 LEDs in series, drive from +12V, and use a current-limiter transistor similar to what you did for the neon bulbs. Just recalculate Re from the formula I posted earlier, and use 10mA and 3,3V. I get something like 260 ohms.
Lastly, since you are running on a 12V automotive supply, be aware that's a very nasty environment with all sorts of voltage spikes (alternator load-dump), dips (engine cranking), and even reverse-polarity (doofus giving your car a jump start has the cables backwards). It's especially bad for older vehicles that have mechanical voltage regulators. Even the method how you wire it will make a difference, depending upon what kinds of high-current devices share the wiring (fans, A/C compressor, starter, alternator, electric windows).