Rule of thumb: maximum 20mA output current of Arduino pin.
Which power do you want to use to generate the 1000V ?
If the Arduino 5V is used, it will be connected to the Arduino, and you need to take care how you design the circuit.
How much current do you need ? Only a static voltage or do you want to output a frequency or switching output ? I assume only a static voltage and very little current.
You need a driver with transistors or mosfet, then a transformer, then probably a voltage multiplier.
You also need special fast switching diodes, since the UF4007 can go to 1000V. Also many capacitors are only up to 1000V.
Do you want to measure the voltage with the Arduino ? To be sure it is 1000V and not somewhere between 200V and 2000V.
Did you know that you can not measure such a low current / high voltage with a voltmeter or multimeter ?
When you connect a voltmeter, the impedance of the voltagemeter (for example 10M) will drop the voltage a lot. After you disconnect the voltmeter, the voltage could go up a few hundred volts without you knowing it.
My advice : Don't do it. You need to be an expert in high voltages. It is dangerous. Perhaps you can buy a high voltage power supply.