So I wonder why staré věc is wrong if Stará refers to a feminine word isn't staré neutral? Is a thing not neutral?
You might think so but věc is actually a feminine noun so the correct answer is "stará věc". You might want to refer to the "Tips and Notes" sections which are quite helpful. If it is any consolation, I also struggle mightily with nouns and genders.
Read the Tips and notes (or any other resource) about the hard and soft declination of Czech adjectives. Poslední is soft, starý is hard and has different nominative ending for each gender.