I'mguessing that it's designed to run off 12v, then the little box steps it down to 5v USB. You could consider installing a DC to DC converter in your box to convert from 36v to 12v, and then use a standard car charger for your phone. You could step the voltage down to 5v directly, but if the DC to DC converter fails, you get 36v in your phone, which would then smoke nicely. The car charger gives you insurance against that. If you wanted to be really neat, you could mount the car charger inside the box as well and either have a retractable lead or mount a small connector on the outside.
These converters work quite well and are good for 3 amps, so you can use them for lights as well.
LM2596HV DCDC Step Down Converter Adjustable + Heatsink | eBay