Evening and Morning in Madrid in early June - What to See/Do

I would stroll El Retiro park, maybe hit one of the museums in the morning, but time will be fleeting with a 2pm departure. MAD is huge and you will need the typical advance time. I'd take a taxi but there is train.
Maybe google for one of those "weekend in Madrid" type of articles and pick out what appeals to you for your amount of time.

Madrid is so big, and while it makes perfect sense to do a quick flyby like this, you also might consider little Alcala de Henares, a college town very near the airport, birthplace of Cervantes. It is perfect for a night, and came to mind just in case you want something less ambitious, more relaxing.

I would suggest that after arriving in Madrid in early evening, you might not have a ton of energy to head into the city for such a short evening, sleep in Madrid city and then head back to the airport the next day. It takes a least 30-40 minutes to get back to the airport either by metro or taxi.

I would suggest getting a hotel reservation at the airport for the night so you will be near the airport the next day for your next day flight. Remember, you will be checking in just like a domestic flight in the US, so you'll need to be at the airport at least a couple hours before if not longer.

After you check-in, take a taxi to Madrid city and have nice dinner and walk around the area near Plaza del Sol and perhaps Plaza Mayor. Taxi back to your hotel when you are done walking around and are ready to crash for the evening. The metro is an option if you are familiar with public transportation otherwise I would stick with a taxi.

Can't help you on vegetarian restaurants, but plenty of sites can assist you with those in the areas around the Plazas. Enjoy.

Evening starts at 6 pm in my mind. With collecting luggage (plus any immigration or customs) plus transport into the city and hotel check in add maybe an hour by taxi. Longer if you take the train. So, you'll be at your hotel by 7:30 pm at the earliest. Still plenty of time to enjoy the evening.

I suggest the Radisson Blu Prado right across from the Prado Museum. I stayed there in 2019 and it worked great for me. They have a good buffet breakfast. You can probably get enough vegetarian breakfast choices.
There are some small outdoor restaurants right in front of the Radisson Blu but I don't know if they offer vegetarian. I had fish there.

The Radisson Blu is a short walk from the Atocha train station if you take a bus or train from the airport. You'll probably grab a taxi to save time.

Across the street from the Radisson is the Royal Botanical Gardens. A bit further east is Retiro Park. It's a nice area on the whole to wander around in.

You depart the next day at 2 pm so have until noon at the very latest to do something interesting. That is assuming you take a taxi. If you take one of the trains, you'll need more time.

The Prado opens at 10 am and is right across the street. The Thyssen-Bornemisza Museum is three blocks away and opens at 10 am. You might be able to allocate an hour to one or the other.

Good luck.

To correct EC, if you have a 2pm flight you should be at the airport 2.5-3 hours in advance. Depending on where the hotel is and how you’re getting to the airport, you can add about an hour to that. That means you need to leave for the airport no later than 10:00-10:30 am.

You won’t have any time to see anything on your departure day if staying in the city, so staying near the airport makes sense.

The Royal Palace is beautiful.

Thanks to all of the above for responses so far!

Anyone have thoughts on Debod Temple at dusk? We may be able to just make it there before sunset.

It’s not a huge site and a lesser site for certain, but you may be able to get some nice photos at dusk. It’s very close to the Royal Palace. Even if you don’t get into the palace its Sabatini Garden is near the temple. If you enjoy Don Quixote and Sancho Panza, the temple is a 10 minutes walk from the small Plaza de España with large sculptures of Cervantes’ characters.