What kind of doctor treats hypoactive delirium, a neurologist or a psychiatrist?

Both, and more: There are consultation-liaison (c-l) psychiatrists working in hospitals, who help medical teams deal with these patients all the time. Also neurologists may do so. The bottom line, though, is to get to the reason for the delirium and treat that. C-l psychiatrists often help with this, discerning illnesses, medications, & side effects that might contribute to the problem and assist recovery.

Depends on your area: Many times both specialties are familiar with treating this but it can depend on where they trained in residency.

Try geriatrician: Geriatrician is very capable of managing delirium. They train with psychiatrist and neurologist during the residency and fellowship. They are able to identify the cause of the delirium and to treat appropriately with proper follow ups or continuum of care.