Wire up your NeoPixel ring as shown: a red wire to PWR, a black to GND, and a white wire to DI (data in).

The massive arcade button comes in two pieces: the large button enclosure and the switch assembly. Remove the assembly from the button (twist and pull) and take a look at it. There are four connection points available: two go to the onboard LED, and the other two go to the switch. The points aren't well marked so I used some alligator clips and a coin cell battery to figure out which were connected to the LED's anode and cathode side. 

Solder a red wire to the anode and a black wire to the cathode. There's already a resistor in there so we can power this LED from any of the power pins on the Sparkle Motion.

Here's a guide showing how these buttons work, if you need more help.

The remaining two pads go to the switch. Solder a green wire to one pad and a black wire to the other. These two are interchangeable (the switch simply connects them together when pressed) but the color coding will help you get everything connected to the Sparkle Motion correctly.

Connect the ring to the Sparkle Motion. Use the first three ports on the screw terminal to connect the LED ring: red to +, black to -, and white to GPIO 19.

Connect the LED wires on the switch to a free + and - terminal: red to + and black to -. 

Connect the switch wires: black to the remaining - terminal, and green to pin 18, along the side of the screw terminal. We can't use the screw terminal data GPIO since the pins they are output-only.

Use hot glue to secure the ring to the underside of the arcade button. Attach the switch assembly to the button, wrap the wires around and use a zip tie to attach the Sparkle Motion board to the switch.

For my setup, I added a right-angle USB C connector, so the cord hangs down instead of sticking out to the side. Your setup will depend on your enclosure.

Finally, I added a layer of black gaffer's tape around the Sparkle Motion board and wires before hot-gluing it into my vase enclosure. The vase has openings in the sides where the light ring's effects can be beautifully seen, but seeing the electronics inside spoils the effect a bit. The black tape makes it all look a little cleaner.