How to preserve this alcohol free herbal bitter?

It is not that it is going to turn bad, the issue is that the interesting components of this bitter are mostly volatile. Volatiles and the flavor components of bitters are more easily "dissolved" into/preserved by alcohol. Water is terrible for this. So, the best you can do is purchase or make small amounts, be sure to keep it tightly sealed, and keep it away from direct light. It's hard to say how long it will last once opened...days maybe, but I can promise it is much less time than a bitter made with alcohol.