For the most part to my knowledge it’s the same as maintaining any large, complex piece of infrastructure. As it gets older spare parts get harder to find and have to be replaced with different similar parts requiring new engineering analysis, more and more big components like pipes and tanks get to the point where they need to be wholly replaced, etc…
Design lifespan is the point the designers expected a lot of annoying to replace things to wear out on paper for the cost of maintenance to rise, but now in the present we can inspect things to see how they actually did in practice.
This means that operations gets more expensive and you need to shut down for major work every now and then, but compared to the ever increasing cost of building an entire new plant just replacing the parts that have worn out in order to squeeze an extra fifteen or twenty years is probably going to be more cost effective to a point.
We just need them to hold in long enough for us to get enough renewables and storage capacity on the grid to replace all the fossil sources, at which point we can keep building renewables and replace the most most expensive to maintain nuclear and most fish limiting dams and the like.
With current battery and hydro storage prices, their cheaper than natural gas with with the cost of the buffer, and absurdly cheap for any industrial application that doesn’t.
Also there are bulk industrial processes to make steel, concrete, fertilizer, and glass with little to no carbon emissions, they just require more electricity and so aren’t cost effective if your electricity comes from fossil fuels, hence why most such plants only started construction once the cost for electricity in general dropped below the cost for fossil electricity.
Moreover while mining and shipping are only starting decarbonization, the required fossil fuel extraction is already far, far smaller than what’s continually required to run the generation they are replacing, and that’s only going to continue to drop as more and more primary energy is electrified with renewables.