Connecting power source to Arduino's 5V pin
Yes, it is correct, as long as you leave Arduino unpowered via 9...12 V DC plug, because it will draw power from that 5V adapter.

And of course, be sure your adapter can handle the current for both Arduino and that Ring.

Another option would be to power Arduino as usually. Power the Ring through the 5V adapter. Then join the ground on all devices (Arduino, Ring, 5V supply).