Tips on touring Prado

Perhaps it will not be as bad as you are expecting. When I went to the Prado in late May 2015, there were knots of people in front of only two paintings, "Las Meninas" and Bosch's "Tree of Life".

I agree with acraven - when I went, the Prado was not empty, but it was not nearly as crowded as the Louvre or the Musee D'Orsay (not to mention the Vatican Museums!).

Do prioritize what you want to see. The Prado is not like some museums, which only have a few great pieces and a lot of second rate stuff. The Prado has room after room of first rate paintings.

As for tickets, I bought mine in advance at the ticket window, toward the end of the day when there was no line. The tickets are date-specific, so do be sure you get them for the day you need.

In terms of other museums that day, it depends on how long you spend at the Prado and how much "museum overload" you will feel. Also, after the Prado, even the other two "biggies" (the Reina Sofia and the Thyssen) will seem second tier. I was very glad I saw each of these on a different day, and saw the Prado last. I realize you said you only have one full free day, so my schedule won't work for you.

We bought our tickets in advance, this was late june this year. It opens at 10am if I recall, so we arrived there at 10, we skipped the regular line, which was not that long at that time and went inside.
We did the museum pretty much in order of room numbers. We spent about 3 hours. Let's say that the last half hour, my concentration and appreciation wasn't there anymore, but we saw most of it and stopped several times at important paintings or painters, reading in my book or commenting.
Personnally, I had enough of museum for one day, though I love paintings. So we went for a long walk in the retiro park behind, beautiful park, a must see. We spent a few hours there, went up to Puerto de Alcala and Placa de cibeles, 2 major outside sites.
After that, my friends were ready to go back to a museum, so they went to Thyssen I believe for a few hours.
I think it's a nice way to cut the day in 2, see 2 museums with a long break walking in a gorgeous park.

I agree with the others, the Prado is so massive and so filled with masterpieces, that you can spend hours there, not see everything, and get major museum fatigue (I too LOVE art museums). And the other museums to seem to pale compared to the Prado - so if doing more than one, try the others first and the Prado last. I also didn't think it was overly crowded like the Louvre, D'Orsay and Vatican museums have become.

In October 2017, we used the Paseo del Arte ticket so we didn't have to book ahead for the "Big Three". We got to the Prado shortly before it opened and eventually found the correct, and shorter, line. The free brochure with the locations of the most famous works is a great guide. We made a beeline to the room with Las Meninas and other paintings by Velasquez. We had them practically to ourselves for several minutes.

You could probably do the Prado, Thyssen and Reina Sofia in one day - making the Paseo del Arte ticket worth the money, plus the flexibility of not having to book ahead. The Thyssen is a great museum and not particularly crowded - except in the gallery with the Impressionist paintings. We picked up the Paseo del Arte ticket there. We really didn't spend that much time at the Reina Sofia - basically tracked down Guernica and "skimmed" the rest of the collection. IMHO, it's worth it for Guernica.

Do the Prado as soon as they open at 10:00. Spend 60-90 mins, 2 hours max to avoid burn out. Then go for a walk in Retiro Park or the Botanical Garden to refresh your mind. As suggested by claudecyr65, go do something else besides museums.
If you still have energy, you can go to another museum before or after dinner.
The Reina Sofia is opened until 9:00 pm. Closed Tuesdays. There is a gallery talk in English about the Guernica on Thursday at 5:00 and Friday at 11:30.

Thyseen-Bornemisza is opened until 7:00 pm on Tuesday-Sunday. Monday is free noon-4:00. I read on TripAdvisor that it is not too crowded on the free day.
Another alternative is to see something other than fine arts is the archaeology museum. Closed Monday. Opened until 8:00 pm Tuesday-Saturday.
If you are over 65, there is a discount for the Prado and T-B, free at Reina Sofia. Need to show your passport.

The Naval Museum is a short walk from the Prado and is worth the 3 Euro suggested donation to see the earliest preserved map of the Americas. I think it takes less then the 6 minutes shown on rome2rio.

https://www.esmadrid.com/en/tourist-information/museo-naval?utm_referrer=https%3A%2F%2Fwww.google.com%2F

https://www.rome2rio.com/map/Museo-Del-Prado/Naval-Museum-of-Madrid

Great ideas/comments! Will only try for Prado as per the suggestions. Sounds like trying to see more in the one day I have will be exhausting.

When I visited (in 2010) the museum had a great flier with all the greatest hits on it - showed you how to find all the big pieces (Bosch, Las Meninas, Fra Angelico, etc.). This page has several maps for 1, 2, 3 hour visits https://www.museodelprado.es/en/visit-the-museum (click on "download plan" to access the full map). Personally, I'd do an abbreviated Prado visit and see the things that interest you. I think the Thyssen was a more manageable and enjoyable art experience. You could do Prado in the morning, have lunch, and then visit the Thyssen (it's very close by). There is a VIPS restaurant near both - if you're tired of eating ham by this point in your tour, they serve pretty good salads.