In your first example with a MPSA42 (NPN BJT), when the arduino switches high (5V), your NPN does turn ON. However the voltage presented to the anode of your nixie tube is only \$5-1V_{be}\$ which is approximately 4.3 V. No good, for a high-voltage nixie tube.
As you can see, to use an NPN transistor as a high-side switch you need to pull the base 1Vbe above the supply rail you want to switch on. Additionally you need to pull the base all the way back to ground to turn it off. Because of these requirements, an NPN is rarely used as a high-side switch.
In your second example, you are using a PNP as a high-side switch. In this case all you need to worry about is current. When you want to switch on the PNP, pull \$ I_c /10 \$ current out the base terminal for good saturation.
To switch off the PNP don't pull any current out of the base of the PNP. The pull-up in parallel with the base & emitter terminals of the PNP is discharge base charge and potential base leakage current.
Note: One resistor is missing from your second schematic.
Sample schematic of a high-side PNP switch for a nixie tube is shown below,

simulate this circuit – Schematic created using CircuitLab
To select components \$R_x\$ & \$ R_2\$ we start from the load and work backwards towards the control signal.
We will assume that max load current of a nixie tube is 20 mA. To ensure good saturation of Q2 we desire a base current of,
$$ I_{B2} = \dfrac{20 \text{mA}}{10} = 2 \text{mA} $$
When Q2 is conducting we know \$V_{B2}\$ will be one \$V_{BE}\$ below the supply rail \$ \approx 169.3 \$ V (170 is also just as good of an approximation here).
The pull-up current through R1 is,
$$ I_P = \dfrac{0.7}{100\text{k}\Omega} = 7 \mu \text{A} $$
The collector current of Q1 is the sum of \$ I_{B2} + I_P \$ which is approximately just \$ I_{B2} \$ = 2 mA.
Resistor \$R_x\$ is a dropper/ballast resistor used to set the base current drive of Q2. We know \$V_{B2} = 169.3\$ V. The collector voltage of Q1 (\$V_{C1}\$) can be approximated as 0 V (typically 200 mV is used as an approximate Vce,sat however its negligible in this circuit).
By ohms law, we can find Rx as,
$$ R_X = \dfrac{ 169.3}{2 \text{mA} } = 84.650 \text{k}\Omega $$
Selecting the next lowest value in EIA24 resistor series would be 82 kOhms.
Select \$R_2\$ for at least 200 uA of base drive. Anything lower than 25 kOhms would be fine for \$R_2\$. Personally, I would just use a 10k resistor for \$R_2\$.