Use the circuit diagram to reference for connecting the components together. The diagram does not depict exact wire lengths or size of components.
If you're planning to leave the lamp on for extended periods of time, it's best if you power the circuit via 5V USB power supply. You can use your computer's USB port, USB hub or dedicated 5V power supply.
Build a Prototype
If this is your first electronics project, it's a good idea make a prototype before mounting the components to the enclosure. You can use alligator clips to make quick connections without having to solder any wires. The idea here is to test the code and components to see if everything works as expected. You can ommit wiring the slide switch to the 500mAh battery in the prototype - Plugging the male JST connector from the battery to the Adafruit Feather board will prompty power the micro-controller on, unplug to power it off.
Alternatively, a breadboard is meant for prototyping, but soldering headers and desoldering headers can be a bit of a challenge.
Once you have uploaded the Arduino sketch to the Adafruit Feather 32u4 Bluefruit LE and test connected the NeoPixel Rings and 500mAh battery, verify everything works. With that, you can proceede to the 3D printing and assembly.