How we can find hot spring genesis time?
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.