Yes, you can put it in the fridge to help keep it fresh. What I usually do to keep the tobacco moist is put in a couple slices of bread into the bag (or drum, or pouch, whichever I have). The tobacco pulls the moisture from the bread when it starts to dry. You can also use this method to rehydrate dried out tobacco that is too dry to roll. Make sure you check the bread to prevent it drying too much and truning into crumbs in your tobacco. Also, you have to keep your bag sealed tight, otherwise any effort you put forth will be for nothing.
For the short term it should be fine. For long term storage I suggest the use of mason jars. I've got tobacco that's nine years old in jars.
As long as its sealed airtight, it should be ok.
You can try keeping it in the fridge, but again, keep it airtight as you dont want it getting wet either.