Can I run arduino on Windows Embedded Compact 2013?
What OS is running on two different devices exchanging messages/signals, does not matter. What matters is the protocol. 

You can use a lot of different ones:


Wireless:


IR (infrared)
RF (radio), like 433 MHz, 2.4 GHz
WIFI (internet)
Bluetooth

Wired


SPI
'simple wires' / Wire library
Serial
USB (not on most Arduinos)



This list is not complete, just some examples.

Arduino supports all above (except USB fully), I don't know what Windows Embedded Compact OS supports.