How do I connect an avalanche photodiode to an Arduino Uno?
You can connect case pin to the ground and that is a safer side too. Normally you will connect the anode of the photodiode to the analog pin of Arduino and cathode to the ground. Cathode and case pin shares the same ground. 



When you see the image 3rd pin is the case pin which has no connection with the photodiode. 
Also, since it is a low light level measurement it is suggested that using a transimpedance amplifier in a circuit is a good choice. 



Based on the bandwidth, noise and sensitivity and other important factors, you should choose an appropriate photodiode for this photodiode.