Trying to pack too much into one day in Madrid??

Are you flying in from the US or Canada that day? In that situation I would definitely not want to spend money on a flamenco show and dinner beginning at 8 PM. However, some travelers manage such feats of endurance. All I accomplished on my arrival day was a visit to the Botanical Garden.

If your trip is a summer one, that 1 PM walk to the Palacio Real will likely be uncomfortably hot, and you might prefer to taxi from the Prado to the palace. If the weather is good, it will be an interesting walk. One possibility for lunch would be the San Miguel Market, where there are a lot of stands offering high-quality tapas. You would likely have to eat them standing up, though, and you might not want to do that on a day when you'll probably be quite tired.

When I arrive from my long flight over, I plan almost nothing for my first day. I do wander around and check things out and maybe, just maybe I figure on doing one lightweight activity. I would not find it fun to fly all night to Spain, drop my luggage, eat breakfast, tour the Prado, walk to the Royal Palace, have lunch, visit the Palace and then get ready for a Flamenco show and dinner. That's a long long list. Have you ever tried do something similar on an arrival day to Europe? If you have and you know your stamina, then maybe it's workable. But you asked if it's a reasonable plan: my thought is no. I would not hesitate to plan a day like this once I'm acclimated though.

Thanks Valerie and acraven! I think you both have validated my concern. My husband and I will likely be back to Spain for more of a land trip but the couple we are traveling with won't so we will most likely try to to fit in the Prado and Royal Palace. I liked the idea of a quick bite to eat at the market for lunch and most likely an early dinner as well even though that might be hard to find. No flamenco this trip! Our hotel is across the street from Retiro Park so we will have a chance the next day to walk around there. We will have four full days in Barcelona so will have a more leisurely approach there.

I forgot to mention that yes, we will be flying overnight from the US and will be visiting in late September so hopefully a little cooler weather than now.

They will most likely feed you breakfast on the plane, so you might not need to take time for that once in town. Remember, too, that "arriving at 8:15" doesn't mean sightseeing by 9:00. You will have to collect luggage, go through immigration, get in from the airport (30 minutes), and get to your hotel. On my recent trip my flight got in at 8:30 and by the time I got into my hotel and ready to head back out again it was past 10:30.

It might be a better use of your time to walk around Retiro Park instead of the Prado after a long flight. Staying out in the sunlight is better than walking around in a museum. If it is walking weather, you could easily walk from the park (or the Prado) through Puerta del Sol and on to the Palace. Mercado De San Miguel does have sit-down tables, or there are numerous little bars with outside tables nearby. There is really no need to book ahead for the Palace. There is a VIPS restaurant in the same building as the Westin Palace hotel (which you will see across the street from the Prado) where you could easily have an early dinner. I wouldn't bother booking a flamenco show, though, because it will run way too late for your first night. You will be tired early.
EDIT - the OP mentioned seeing Retiro Park the second day, but I would still think you would be better off to do the Prado when you are fresher.

Another thought - would your friends be interested in a museum that isn't fine arts? A block from the Prado is the Naval Museum, very interesting with models of 15th-18 century ships, maps, etc. It would be easy enough to split up and meet 2-2.5 hours later. Also, consider visiting the smaller, more manageable Thyssen Museum which focuses on 18th-20th c. painting - it's a block in the other direction from the Naval Museum.

After a long-haul flight arriving early morning, I wouldn't have the stamina for afternoon sightseeing. I'd probably choose to nap in the afternoon and then go out in the evening. I certainly wouldn't be able to go like an energizer bunny for 15-16 hours.

The market will be open for an early tapas dinner if you are hungry after you see the Palacio Real and head back east--which you might be if you've only nibbled at lunchtime. It's not easy to find a sit-down dinner in Spain on an American schedule, and you'll almost certainly want to eat and crash very early that first night. If you're really early, the market will be quiet (not much of an opportunity to experience the tapas scene), but the food is good, and the great variety will work well if the four of you have different tastes.

As, I said, VIPS (overlooking the Neptune fountain, and near the OP's hotel) has a full sit-down menu and serves dinner at an "American" time. If I eat early, I am never alone in there - plenty of others eat early, too. They have, for one thing, the best club sandwich anywhere.

I thank you all for your thoughts! Based on the replies I may change our itinerary. We will plan to visit the Prado the afternoon of our arrival after spending time walking through the park, getting a bite to eat in the market, etc. This plan is a bit more leisurely. We may try VIPS since there doesn't appear to be many options although it gets very mixed reviews. Our hotel has a bar serving food so that is also an option. The flamenco is definitely off the agenda! We will take the train to Barcelona later the next day than originally planned and visit the Royal Palace when it opens at 10:00 (although I will be disappointed at not seeing the Caravaggio exhibit which ends the day before.)

Start at the Royal Palace and (depending on your energy levels) finish at the Prado, where the last two hours are free.
Royal Palace >>Almudena Cathedral >> Plaza Mayor >> Puerta del Sol >> Gran Via >> Plaza Cibeles >> Puerta de Alcala >> Prado >> El Retiro Park
Prado (€14) Tue-Sat 1000-2000, Sun 1000-1900, Closed Mon (Free after: 1800 Tue-Sat, 1700 Sun)

If you enjoy museums and your travel companions don't then split up for this day and spend more time in the Prado and let them go do something else. Nothing makes one tired of travel companions faster than being forced to do things you don't want to do with them. You have the whole cruise with them. Personally I would not bother with the palace which is not particular interesting and the Cathedral is equally uninteresting. The Prado on the other hand is amazing.

Like others, I would be falling asleep in the soup if booking a late dinner on a day after I had flown from North America and then spent a day rushing around Madrid. YMMV.

Consider your doing the Prado longer and them doing the Palace and then meat to tour perhaps this lovely little church that you used to be the Cathedral before the current unimpressive one was built.

https://janettravels.wordpress.com/2013/10/23/easter-egg-in-madrid-basilica-san-francisco-el-grande/ You could do a tapas crawl on the nearby Cava Baja and after an after dinner drink near your hotel crash and be ready for the next phase of the trip the next morning.

I agree that the Basilica San Francisco is a semi-hidden gem, including its own mini pinacoteca.

It is also an important spot for the suppressed history of the Basque independence movement--
on the curb outside the front entrance was where an ETA car-bombing incident assassinated Luis Carrero Blanco as he left Mass a few days before Christmas in 1973. Reprisals by Franco followed, and heated up the conflict.

Another point-of-view on dealing with the jet lag: use it as a chance to get on local patterns, by taking a siesta in the afternoon and then going out for dinner after 9pm, the more civilized hour. To me, those restaurants and cafes near the Prado that serve non-Spanish dishes at non-Spanish hours are the height of gaucherie.

The "local pattern" of Madrid does not include a siesta. All businesses are open all day. Your most recent plan sounds like the most workable to me.