"قَلَم اَلْأُسْتاذة بُنِّيّ."
Translation:The professor's pen is brown.
Does this mean there is no way to mark possessive for 3rd person full nouns? ... Also, how would you say "a professor's pen"?
a professor's pen: قلمُ أستاذٍ
if you by 3rd person possessive like (his pen, her pen) then it is possible of course. That would be (قلمُهُ, قلمُها) respectively (respectively from right to left).
قلم الأستاذة البني (AL is added to the adjective بني)
In this case the adjective is said to be "attributive" and hence it follows the noun it describes in status (number, gender and definition by AL)