Is there any way we can calculate the time when a particular hot spring come into existence? Also I wanted to know what is average time period for a active hot spring in general, whether it can flow for geological time scale or can be only in 100s of years?
The answer to the first question is a maybe, hot-springs often have mineral saturated water that can create layered deposits which may be datable, much like tree rings. If reasonably uniform flow and mineralisation rates can be demonstrated over time then reasonably accurate dating may be possible.
As to the second question, we have a saying "how long is a piece of string?" meaning there is no universal answer, some springs may be seasonal, only flowing for a few months at any given time, while others may flow continuously for hundreds of years then stop abruptly for no apparent reason. There are no "in general" each and every set of geothermal plumbing is unique.