Will the price become more steady or will it continue to fluctuate as it does now?
This will happen in year 2140 or so.
To make a prediction for something which is that far away is almost impossible.
However if Bitcoin would still exist around that I assume it achieved to make the step of being a world currency.This would mean world adoption would have happened and billions of peole are using it.So the volatility wouldn't be as much as now, more the opposite would be the case.And transaction fees would take over the place for reward as there would be many millions of it each day.
Something like that would be a very positive any maybe naive to believe future scenario.
The bad scenario would be that Bitcoin disappeared and something else is in it's position.Therefore more explanations of what will happen regard the last mined BTC is not needed.