Absolutely carry on and you should drain the batterys before taking on a plane. If you feel the styrofoam case is good enough then by all means use it, it's what the whole package ships in so you can consider it safe, eventually you will outgrow that case with accessories, batteries, upgraded antenna etc.
The case does not come with any type of external straps so will have to be carried by hand which will also eventually annoy you, especially whilst travelling. In my opinion you should use the original case for now until you outgrow it and decide which accessories and how many batteries you need etc by that time P4 cases will be put through their trials and get some more reviews, also prices may settle down a bit. Also here is what seems to be turning into the P4 backpack thread
New Phantom 4 Backpack