I forget all the choices I made to make this happen, but as the Critter, go into the bar and find a book for drink options. Page through it and, at the third page (or so), find the "hangover remedy" drink.
It requires a number of items you find in the island. You will need peppermint leaves (get these by ordering a different drink from the bartender that has peppermint in it ... save the leaves.
Then get a pan from a statue (for get if this is in the palace or the town square) and put water in it from the palace. Then head to the piers and set the pan on the bitt. Then boil off the water using the nearby mirror to give you salt. Save the salt.
After you leave the palace a coconut will drop from a tree ... get it. When you're in the palace, use the Sword in the Stone to cut open the coconut.
Now, combine all three in your inventory and finally make Nate drink it. Nate will now be sober.