How do I open my wallet in iPhone x

The Wallet App can be opened by tapping on the App itself just like any other app.

If you want to quickly access the App to use a Pass or Credit card, you can double click on the side button (sleep/power).

More details here:

Use Wallet on your iPhone or iPod touch - Apple Support

What happens when you press it then?

If it turns off the screen, then you are clicking it too slowly.