There are several ways to go. Here is one way:
If I were just setting up a test I would use an "instant on" solid state relay. This is the type that turns on as soon as a control signal is applied.
Any microcontroller could be used to do the timing. Use an opto-isolator to generate a logic signal to the microcontroller. Using a timer, triggered by the GPIO input or just a "Delayms" or similar, turn on the signal to the SSR at some time after the zero cross. Look at
https://www.gavazzionline.com/ for some SSR examples. Your inrush current might be very high, depending on your power supply design so that is probably a more important specification than the average/RMS rating of the device. Don't forget to heatsink it.
On the web, Look for "opto-isolated zero cross detector" or similar for the zero cross detection. There are two main types, one is sensitive only for one direction of zero crossing and the other for both directions. You would buy the opto type you need. In the case of your power supply, without knowing your exact circuit, I assume it really shouldn't matter the polarity so I suggest a an opto-isolator with inverse-parallel leds in its input side. If polarity does matter to you, an opto with a single LED with an external anti-parallel diode will work well. You will need to provide resistors to drop the voltage/current. Eg. Say 10mA is desired. R = 480*1.414/0.01 = 48 k ohms. Power rating: 5 Watts, minimum. Another way would be to incorporate a small transformer, say 480V to 24V to get the voltage down some, then a smaller value/power resistor will be needed. This won't turn exactly at zero cross but it will be a small delay afterwards. However, that delay is fairly predictable unless your AC waveform is very distorted.
So, 480V. You need to take great care when working with a voltage like that. You might walk away from a 120 or 240V shock but with 480V it is less likely. Be sure to choose an opto-isolator with the proper isolation voltage. Keep your HV and LV physically separated and isolated. Cover the HV side with something so you CAN'T touch it. I suffered a 480V shock when I was a young pup. It was unpleasant to say the least and I'm lucky to be here to talk about it.
I personally would avoid the PC / Labjack in order to keep things simpler but you could use it if you prefer.