Only way I know is ice cores. But I doubt the precision claimed. There may be other means they do in tandem but I don’t know them specifically.
Discussion
They study fossilized plankton. It is possible that the acidification can be adapted to quickly enough or that plankton that can handle it flourishes and helps to bring down the acid levels. We really don’t know.
The issue isn’t that the oceans can’t handle more co2 it’s that the life in the ocean may not be able to handle the rapid influx of co2 and moves in ph. The ph scale is logarithmic so minimal changes could mean a mass die off of critical animals.