That cabinet looks awesome.
Applying weatherstripping is not as easy as it sounds. You have to still have the door close properly. The easiest solution usually is to move the hinges out so the doors are mounted an even distance out from the cabinet all around. Lots of strong magnets are needed to hold the door closed with even pressure. I would say about 3-4 down the height of those doors. Also, I would suggest sealing all the back joints with odour-free silicone caulk to improve the seal.
You may also want to drill some holes or cut slots in the shelves to improve air circulation. A 1" hole saw would make quick work of it.
