Disclaimer: Lately, I was checking some unanswered questions. This one stands for a while. So, I decided to use the advantages of sitting in an academic office and sought the academic databases I can think of. Some of them are only accessible by abstract. However, I will try to copy the essence of them here without disturbing copyright rules of the publishers.
First things first, I tried to understand the effect of going stale to the amount of antioxidants in coffee. It seems not.
My first reference is "Antioxidant capacity, phenolic acids and caffeine contents of some commercial coffees available on the Romanian market." by Trandafir I., Nour V, and Ionica M.E. which you can read its abstract here. According to this article, total amount of antioxidants in a regular coffee sold in the market is changing in between 19.8 to 41.9 mg per gram coffee.
My second reference is "Recovery of Natural Antioxidants from Spent Coffee Grounds" by Alessia Panusa, Antonio Zuorro, Roberto Lavecchia, Giancarlo Marrosu, and Rita Petrucci which you can read its abstract here. According to this article, total amount of antioxidants in the remains of the spent espresso waste is changing in between 17.07 to 35.52 mg per gram coffee.
I conclude that, even after brewing (chemically speaking; extraction or leaching) the antioxidant amount is not changing much.
Then, I tried to understand what changes the antioxidant level in coffee, if there exist something. Yes, there is one; here comes the third reference by Ayelign and Sabally, "Determination of Chlorogenic Acids (CGA) in Coffee Beans using HPLC" which you can download a copy of it from here. In this study, it is shown that the level of CGAs (an antioxidant) is dropped from 46.144 mg to 0.981 mg per gram coffee, more the coffee is roasted.