I'll be going on vacation for 1+ month and leaving my Prius C parked outside for around 30-40 days. Is there any special precautions I should take to make sure that my Prius will start successfully when I get back?
For 30-40 days you really don't need to do anything special. Unless your 12V battery is nearing the end of its life, which it should not be yet.
If it is going to be outside, I would use a sunshade and other things to protect the interior from heat. The other thing would be to watch out for rodents