eos seems to be mostly shea butter and jojoba oil. The trick is to find some solvent that will dissolve those things but leave the finish on your table. Since I don't know what air hockey tables are finished with, I can't say for sure about that.
As total guess, you might try some dish washing liquid straight out of the bottle, let it sit for a while, rub a little, then clean it up with a sponge and water. If you're lucky the eos will come off with it.
If that doesn't work, try rubbing alcohol, but not a lot at once.
WD-40 might also be worth an experiment. It's surprisingly good at cleaning up greasy stuff.
Organic solvents like toluene (available where house paint is sold) will almost certainly take off the eos, but might well damage your table too. (On the other hand your table is already damaged, so maybe try just a little at first with a cotton swab to see how your table reacts.)
You might have to clear each air hole individually with a toothpick.