Answer: There are two answers to your question, yes and no! Yes you can grow a grape in a container, although it would need to be at least the size of a half whiskey barrel and be watered faithfully. But, no those varieties are not suited to Austin's climate.
Table grapes in general suffer in the central Texas climate, and none are recommended as dependable by the Extension Service. Several wine and jelly varieties are feasible however, such as the tough, proven grape 'Champanel'. For more information, check out the following web page:
I might also add that I would suggest you plant them in the ground (if that is possible) and train the vine up to the balcony level. This way they will have much better vigor and resilience. I saw this done extensively in Europe and was quite impressed with their results. You could even place a large container on the ground knowing the roots would go through the hole in the bottom and establish themselves in the soil. Then, if you moved, you could cut the roots emerging from the hole, cut back the vine, and transport your plant.
Thanks for the question!
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