You can use pretty much any box you want for your reverse geocache. Make sure you don’t cover up the GPS antenna with any metal that would block the reception. I made my box out of cherry and cut holes for the LCD and Power Button. I cut matching 45° angles on the back side to allow the box to open 90°.
Next, insert all of your electronics. I cut a square hole in the side of the box with a jigsaw so I can connect to the Uno without taking it out of the box. I connected the circuit up to the power switch and mounted the servo on some standoffs.
I added some foam covered in velvet cloth on the bottom to hold the ring:
I added a latch mechanism to the servo arm that locks the box (not shown).