I will begin this by stating that I am a PhD Student in potato post-harvest physiology and the sprouting of potatoes is an aspect that many people that work with this crop find very important.
It is certainly feasible that the potatoes, if refrigerated, would be able to sprout after several months. Some potatoes, including modern varieties, are able to sprout 2 years after their harvest if kept in a refrigerated environment. What is interesting about raw potatoes is that they are biologically alive and current preservation techniques aim to modify their urge to send sprouts to grow into a new plant. Other preservations techniques beyond refrigeration would include sprout inhibitors, chemicals that inhibit hormone mediated pathways that cause sprouting and package membrane technology. I imagine the packaging in this sci-fi movie is one that retards respiration and lengthens the life of the tubers.
However, the biggest issue with the plot resting on potato propagation is not the initial sprouting and growth. Rather, it is the rapid propagation of a second crop immediately after harvesting the first. Nearly every potato cultivar found in Europe and North America has a period of dormancy after harvest. This means that the tuber itself has hormonal controls that prevent any sprouting for a given period of time. While there are ways to overcome this, even with chemical sprout inducers, there is still a 1-2 month time period where the tubers from the previous crop will not sprout. Barring any chemicals to induce sprouting, it would take 3-4 months until the harvested tubers would sprout into a new crop.
Wiltshire, J. J. J., and A. H. Cobb. "A review of the physiology of potato tuber dormancy." Annals of Applied Biology 129.3 (1996): 553-569.