Retrospectacle has a great post that describes how the Ancient Egyptians removed the brains of the dead before mummification and notes some of their neurological knowledge.
The Ancient Egyptians described a range of neurological and psychiatric disorders in their writing that would be recognisable today.
One major source is the Edwin Smith papyrus, another is the Ebers papyrus which has quite a significant section on psychiatric disorders, including what we would now class as depression, dementia and psychosis.
Needless to say, the remedies were often magical in nature, but the observation of the clinical features can be quite astute.
The article on Retrospectacle has some great brain scan images and a link to a video of how embalmers would remove the brain through the nose, using a metal tool to go up into the frontal lobes.
Link to Retrospectacle article.