This is actually quite a good question, and I think one you can only answer it properly if you have actually tried it yourself, so it would be nice if you can post your findings in a months time :-).
Here is my take on this, and also some considerations:
I would actually try to vacuum pack it rather than using an airtight container. There are always some air in an airtight container which can spoil food.
Also, I would rather make enough for a week, a month is quite a long time to store something like this. I believe it will last for a month if properly vacuum sealed and stored correctly, and if it does not get openend regulary. The only concern would be if it will keep its spicyness for so long. Vacuum sealed or not, frozen or not, food still breaks down at a certain rate. You can't stop it, but you can slow the rate of deterioration down.
Another issue that I see here is that it would also become another hassle of continueing opening and re-sealing your stock. This might and will most probably lead to that by the end of week 2 or 3 that your wasabi has lost its spicyness or even have spoiled to some degree. My advice would be to split your paste in say maybe 4 portions, a portion per week, and sealing them separately. You can even decide to split them up into smaller portions. I really do think that the key here will be to not open, seal, re-open and resealing your stock.