This method does work but can be rather dangerous - mainly because it is hard to judge the carbonation of the "soda" and the glass bottle might become over pressurised. This danger can be alleviated using some basic precautions. It would be far better to use plastic PET (fizzy drinks) bottles than glass bottles. The reason being that one can test the pressure of the bottle by giving it a squeeze. That and the fact that "exploding" plastic bottles are not nearly as dangerous.
The easiest way to make cheap soda is by using a measured amount of sugar with the yeast. I generally use about 150g of sugar to carbonate 20L of beer. I use 350-400g of sugar to carbonate 20L of fizzy wine (champagne method). Using these base measurements one might make up a solution of (say) 10g sugar/glucose in 1L of water. If the juice already has this amount of sugar present then it might be directly fermented - but be warned most juices contain more than this. If it has more then it may over ferment and might need dilution to be used "safely", if it has less then one can add sugar to the brew to achieve carbonation.
Alternative one could make up the required sugar water base and flavour it with syrup or fruit squash- but again check the sugar content of the squash and adjust as needed. Fruit juice with little sugar (eg lemon juice) would add big flavour but little to the carbonation process and so is an "ideal" flavouring agent.
Another possible flavouring agent is the use of home made jam. It contains sugar and fruit flavour so is a useful base for fermentation.
As the yeast will ferment most sugars it can be difficult to predictably and stably sweeten the drink. To do this easily one can use a non-fermentable sweetener. Some prefer lactose or artificial sweeteners but I have found that stevia liquid is a good, low cost option.