Milan for just one day

Book a timed ticket for da Vinci’s “The Last Supper” at Sta Maria delle Grazie.

I don't see anything about Wednesday closures on the Duomo's website. Both the interior and the rooftop are wonderful, though I admit that the tickets are not cheap at €20. You can save some money (paying only €15) by climbing the stairs rather than taking the elevator to the roof. I don't know how many steps there are, but it is a very large building.

You can walk through the beautiful, historic Vittorio Emanuelle II Galleria shopping arcade and the historic Brera District for free.

I believe this is the official website for Last Supper tickets, though they are not selling tickets currently. Perhaps the venue is closed at the moment for safety reasons. https://www.cenacolo.it/the-last-supper/

The Last Supper is one of Europe's toughest tickets. Posters on this website have spoken of getting up in the middle of the night so that could attempt a purchase the moment tickets go on sale for the date of their hoped-for visit. If you want to see the painting, stay on top of the ticket situation by checking the website very, very frequently. Otherwise, you'll have to pay substantially more for a commercial tour that includes the painting.

Hi -
According to the cathedral's website, the main body of the church is open every day from 9am-7pm. You do need to purchase a timed-entry ticket but they are not expensive: €5,50 - € 6.00 per adult. As you only have one day, buying in advance for the entry time you desire will help you to plan the rest of your day, eh? Here is the info page on the cathedral's website:

https://www.duomomilano.it/en/buy-tickets/

Ignore the bit about "we select the best available seats." That applies to attractions, such as concerts, with assigned seating. Tickets to see additional parts of the complex do cost more, and it's the duomo museum - Church of St. Gottardo in Corte which is closed on Wednesdays.

We also enjoyed a visit to Basilica di Sant'Ambrogio: a very old and interesting church which helpfully provided some signage in English. I do suggest doing some reading up in advance. The website is only in Italian but using google translate, here's the visiting info; entrance is free:

http://www.basilicasantambrogio.it

"The Basilica remains open at the usual times. However, due to the current health emergency and the provisions issued by the President of the Lombardy Region, group visits with or without a guide are suspended until further notice. Bookshop and Museum are closed with the same reasons and methods.

TIMETABLES
From Monday to Saturday from 10.00 to 12.00 and from 14.30 to 18.00

Sunday from 15.00 to 17.00

These are not the opening hours, but those suggested for visits to the basilica, which is open every day from 8.00 to 12.00 and from 15.00 to 19.00. On Sundays, for visits, it is asked to verify that there are no liturgies in progress."

Another chiesa well worth seeing is Basilica San Lorenzo Maggiore; the oldest church in Milan. The website is also only in Italian but if unwilling to use a translation tool, there is a lot of background history you can find with just a google.

http://www.sanlorenzomaggiore.com

Website states it's open MON - FRI - 08: 00 -18: 30, Entry is free although there was a small fee ( € 2) to access 4th- century Capella di Sant'Aquilino. As with Sant'Ambrogio, I highly suggest doing some reading up before you go.

Editing to add: as I believe your trip isn't until July, information could change so I would keep an eye on the websites for any updates which might apply.

The Duomo entry fee Kathy is citing does not admit you to the rooftop, where my mother and I were really impressed by the stonework. (I'm not one for paying a lot of money to look down on rooftops.) It certainly would be a much less expensive visit if you only cared about the church interior.

As I'd said, tickets which include more than just the main, interior body of the church cost more. I'd specifically mentioned the lower-priced ticket as you'd mentioned that you were on a budget. :O)

Just across the piazza from the duomo is elegant Galleria Vittorio Emanuele II. It's a historic shopping mall (built between 1865 and 1877) unlike any you've likely ever seen: an architectural treasure free for a walkabout!

https://www.yesmilano.it/en/see-and-do/venues/galleria-vittorio-emanuele-ii
https://en.wikipedia.org/wiki/Galleria_Vittorio_Emanuele_II

If you could see the Last Supper--that would be great. The Last Supper is nowhere near the Duomo. Maybe hire a private tour guide that can take you to the Last Supper and Duomo. I understand there are budgetary constraints, but time is money and having a guide can give you a higher probability of a better experience. This could be the only time you're in Milan. Many people regret what they didn't spend money on, not what they spent.

In Milan there are even two very important galleries: Pinacoteca Ambrosiana https://www.ambrosiana.it/ and Pinacoteca di Brera https://pinacotecabrera.org/ . Both worth a visit.
You can visit even the Scala Theatre and its museum: https://www.museoscala.org/ .
Of course a visit of the Castello Sforzesco is another must for a day in Milan: https://www.milanocastello.it/

Recommend a guided walking tour. I think we used City Wonders Tour and it was 3 hours. The price included tickets to the Last Supper and Duomo plus many of the other sites in the city. It was well worth the price and we had the afternoon to continue on our own.

We have flown in and out of Milan several times on our way to other places in Italy, so we have spent several days and half-days there. You did not mention art, which is usually my most important thing to see anywhere we go, so besides that our favorite things to do in Milan are the roof of the Duomo (not the inside) and the Monumental Cemetery (which is free to visit). There are other large old cemeteries full of interesting sculptures and monuments (and we have been to half a dozen of them), but the one in Milan is the most over-the-top and melodramatic.

Just look at Google images for Cimitero Monumentale Milano sculptures to see if you might like this sort of thing.

Definitely try to score a ticket for The Last Supper and a visit to the Duomo is also time well spent. We had fun just wandering around central Milan and have been there twice for one-nighters similar to what you are doing.