"port" is greyed out in 1.6.13, running win 10 for an Uno R3
When the port is greyed out its not available (I know you had worked that out) so you have a problem with you serial to USB drivers.

Looking at the kit I think you are connecting the UNO to a USB port on your PC/laptop using a blue cable.
 1. Please can you check that the cable is connected properly.
 2. Can you try a different USB port on the PC (don't use a USB hub - some don't work).
 3. Does Windows recognise a device has been plugged?
 4. Are you running it as administrator, you will need to be to install the drivers.
 5. Can you try connecting it to a different PC?

If these don't fix it then, please comment and well go through looking at the device manager.

Hope it works, it looks like fun.