Train from Madrid to Cordoba - advice on timing after overnight flight?

Since there are many options, I probably wouldn’t book in advance. You can buy your tickets online using the Trainline app. This is often easier for foreigners than the Renfe app, which often has issues processing foreign credit cards.

The 8am train works if you are prepared to take a €30 official taxi from the airport to the train station or you book a Renfe ticket with an “ELiGE” supplement.

You can buy the least expensive non-refundable and unexchangeable Renfe train ticket and the ELIGE supplement transforms the ticket into a flexible ticket for an additiinal €8 or so. The ELIGE option allows you to change the ticket to another train—departing earlier or later—ON THE SAME DAY as the train you hold a ticket for— at no additional charge. Unlike the way it is with most train tickets—you don’t have to pay the difference between the price paid for the original ticket and the price of the new ticket The new train you want to take must have seats available in order to change the ticket. ( And you will definitely want to have a seat on a train traveling 186 miles per hour)!
You have to change the ticket at least 30 minutes before your original train’s departure time. . But since it can be done both online and at the airport’s train ticket counter— changing the ticket is quite speedy. So, after you land at the airport, collect your luggage and go through passport control— you can then make a decision on keeping your original ticket or changing it.
IRYO uses the sleek “ Red Arrow” Frecciarosa trains that Italy’s Trenitalia first launched on its high-speed routes. If IRYO’s train schedule has a train leaving about 8:30, i I wouldn’t hesitate booking it—2.5 hours should be plenty of time from your flight’s scheduled arrival time until a 9am train departs Madrid’s Puerta de Atocha train station. In a pinch, an official taxi from MAD Airport to Madrid Atocha Puerta de Atocha train station costs €30 from the official taxi line at the airport.
If you book a Renfe or Iryo high-speed train you are allowed to travel free on Cercanias trains if you get your ticket from one of the machines at the Cercanias station.
The Cercanias rail station is underneath Terminal 4. Look for the sign that points underground and says “Aeropuerto T4.” There are free buses at the other terminals that will take you to T4 in 15 minutes. Buses run every 15 minutes and the Cercanias train takes 33 minutes to reach Puerto de Atocha train station. Allow a generous 30 minutes to get from the Cercanias train arrival area to walk to Puerta de Atocha where you will need to push your luggage through the x-ray machine at the security checkpoint. Only your belongings are x-rayed and the checkpoint is just a few hundred feet from where all high-speed trains depart.

Have fun in Spain!

www.TheTrainline.com
www.Renfe.com
www.Iryo.eu/en

Just in case you decide not to buy your Cordoba ticket in advance: There's a train station at the airport, in Terminal T4. You can buy tickets there. There was no line for me in 2016, and I don't recall reading reports from others who encountered a line.

If you buy your ticket in advance it covers the fare from T4 to Atocha.
We allowed 3.5 hours and ended up waiting an extra 1.5 hours at the train station. We did not check bags and everything ran smoothly and quickly though.

We too have used the the train station at T4. Full service and no crowd. We were able to get tickets to Toledo and purchase Tarjeta Dorada card. Much easier than dealing with the chaos at Atocha. Brad

A certain (higher priced) class on the IRYO train allows free entrance to the waiting room at the ONLY YOU HOTEL across the street from Atocha station. But the waiting area opens at 9:30am. This ticket also allows you to change your ticket for an earlier train, which I did last March because I did not fancy waiting around for my train's departure time. The IRYO site explains it all. There are options on IRYO for meals, and for assistance for travelers who require help with lugggage, etc.