Chakal your friendly Mage Scribe here. Let me take you on a brief tour of Cair Paravel.
First off all of Narnia is for the Talking Animals. The purpose of the Kings and Queens is not to dominate but to serve. The Old Heidelberg doctrine, you know, that to serve is the highest calling and the greatest ruler is the one that is servant of all.
With that in mind, given the tiny human population (four during the golden era) no, Cair Paravel was not for humans alone. It was not built by the hands of man, and men did not paint it, clean its streets, argue in its legislative chamber, defend its walls, cook its food, store its larder, or hoist its flag each morning.
Government in Narnia is locally grown. There is not a lot of infrastructure compared to classic European monarchies. The High King presides over a Privy Council, his advisory group, which is made up of two branches: 1) the Council of Ministers (persons over various functions such as defense, trade and standards) and the Royal Magi, of which your humble servant is a member.
The Chamberlain is the main inkeep who runs the daily affairs of the household and takes personal responsibility for everything from keeping the lawn tidy to repairing leaky roofs. He has a huge staff that is run, by all accounts, like clockwork. And speaking of clockwork he maintains the water clock in the courtyard.
The Captain of the Guard has four shifts of brave souls that maintain order, give directions, and in times of war kick some pretty mean hindquarters.
The Stablemaster is in charge of the staff of grooms that tend the horses and the beasts of burden.
The Order of Saint Suleyman is a lot like the Knights Hospitaler in your world...they run a ward for the sick and injured that is second to none.
Various Trade Guilds have their offices at Cair Paravel. There is also a school of Natural Philosophy and a school of Arts and Music.