Colder temperatures increase the solubility of carbon dioxide in the oceans. The carbon in the atmosphere has to go somewhere. It can go into the oceans with cooler temperatures.
It's not about solubility. Read reference 10 in point #5 of my profile. (It's an abstract, it's short). It's about ocean circulation, ocean mixing, enhanced weathering... basically a whole lot of things. The Eocene-Oligocene transition was an even bigger shift. There had to be increased deposition of sedimentary carbonates, perhaps more calcification; as the article alludes to, even changes in the terrestrial biosphere. The CO2 went a lot of places.
The question is why did the temperature drop?
Because of a drastic change in the radiative absorption properties of the atmosphere -- that's why the ran the climate models!!
Milankovitch Cycles,
Not nearly enough. The Milankovitch cycles can initiate changes, but they don't cause enough of an incoming energy change to sustain them.
decreased sunspot activity like that being observed now,
Little Ice Age, maybe. For an Eocene to Oligocene drop, the Sun would actually have to shrink. There's no evidence that it did.
increased volcanic activity spewing sulphates in the air
Easy signature -- not found. The signature is similar to the asteroid signature -- also not found in the Eocene. The Eocene cooling happened over the whole period. It wasn't a short event like the K/T boundary or a couple of other abrupt cooling or warming events. Big excursions have geochemical signatures.
or a catastrophic impact with an object from space putting a massive amount of dust in the atmosphere.
Because you haven't found some evidence doesn't mean such evidence does not exist. How do you explain the pattern of increased atmospheric CO2 hundreds of years after ice ages receded during interglacials? Nobody was burning anything. What's your explanation for ice ages?