I'm pretty uncomfortable with the idea of just refrigerating bottles of soda that have live yeast in them and plenty of sugar. It just seems like you're destined to some exploding bottles in the fridge.
If I was going to make that recipe, I'd cut everything in half and put it in a 2L soda bottle. Less dangerous if it does explode, and you can easily vent it if it gets too hard.
I don't know how much like "grape soda" that's going to taste - artificial grape flavor has a very different flavor than grape juice does. If you're looking for a more authentically grape, it's probably fine.
I'd skip the boiling, fill a 2L bottle with half grape juice, half water, 1/4 to 1/2 cup of sugar (you can taste it to see how sweet you want it), and 1/4 teaspoon of yeast. Consider adding a little lemon juice if you feel like it needs a little bite, but keep in mind that carbonation will provide some bite as well.
If you want more grape flavor, or a more "soda-y" grape flavor, supplement the juice with some grape extract.