Hold the strip up to your hat's brim to measure the length for the first piece. The design is up to you!
Cut three pieces to the desired length, using pins or masking tape to mock up your design on the hat.
To solder wires to NeoPixel strip, identify the input end (arrows facing away). First solder wires to + and -, then solder the signal wire on the opposite side of the flex PCB. This way your solder joints will be less likely to crowd each other.
Repeat for the remaining pieces of NeoPixel strip in your design, soldering three wires to each input side.
This is a tricky technique that can take practice to achieve a tidy result. You may want to test it out on scrap pieces of strip before soldering the pieces meant for your hat!
Likewise make the connection at the top of the cap, and stitch down any loose wires with a bit of plain thread.