How can I tell if homemade ginger syrup has gone bad?
How long this will last depends largely on the ratio of sugar to water in the final syrup.  Sugar can inhibit the growth of microbes by reducing the water activity of the solution, but this is dependent on the amount of sugar.  According to a book that I consider quite reliable on these matters, a syrup composed of equal parts sugar and water (by mass, not volume!) will generally keep unspoiled for about 2 weeks.  A syrup with 2 parts sugar to 1 part water will keep as long as 3 months.  These both assume that you're keeping the syrup refrigerated in an airtight container, which it sounds like you are.  Unfortunately, unless you know the exact ratio, it's tough to predict exactly how long this will last.

Discard immediately if you notice any of the following:


Any visible mold (hopefully this one's obvious)
A cloudier appearance (harder to judge since you have some suspended particles; you might try running it through a fine-mesh strainer)
Any gas bubbles or outgassing when you open the container (this is a sign of fermentation)


You might also consider adding a small amount of alcohol.  This provides another hurdle to microbial growth, and would extend your syrup's shelf life to somewhere between 3-6 months.  I generally add about 1/2 oz of 80-proof (40% ABV) vodka per 8 oz or so of syrup.

Regarding a mason jar, the main advantage these have is that they won't trap flavors the way that plastic can when re-used.  A strongly-flavored substance stored in plastic for a time can contribute some of its flavor to the next.  Since you're storing a strongly-flavored ginger syrup in a former soda water bottle (which is nearly tasteless) this probably isn't a concern, but if you used the bottle again whatever you put in it might come out tasting subtly of ginger.  Glass doesn't have this problem.