Other than the USB port, there are no external electrical signal connections available to or from the Arduboy. If you wanted to open the Arduboy, there are pads available for all the pins that you could solder wires to and connect them to appropriate pins on the Uno.
The Arduboy and Uno USB ports both can only act as a device, not a host, so you couldn’t connect them directly to each other.
If you only need to send data from the Arduboy to the Uno (not from the Uno to the Arduboy), one interesting idea would be to use light or sound. You could modulate the Arduboy’s RGB LED or speaker and have the signal picked up by an optical receiver or microphone attached to the Uno.
A single direction link might be sufficient. You could select any or all the settings using the Arduboy and then “blast” them all at once to the Uno. You could have the Uno flash an LED or otherwise provide some confirmation that the data was received successfully. The communication protocol you use would have to include a checksum or some other way of verifying that the received data is correct.
Actually, using light there’s a chance that you could both send and receive by using the Arduboy’s RGB LED as both a transmitter and receiver. See my ArduboyLightMeter sketch for the basics on how receiving may be possible.