The best explanation I saw somewhere is to think of the opposites: the opposite of eski is "new", the opposite of yaşlı is "young".

I thought the difference betwen eski and yaşlı was that eski was for things and yaşlı was for people, so how can the müdür here be both eski AND yaşlı?

Is there another word that would be translated as "elderly" or could I use yasli for elderly as well?