Will be in Milan in May looking for hotel recommendations. My budget is $300 a night.
I’ve stayed twice including last Fall at Hotel Gran Duca di York near the Milano Duomo. I didn’t look at the website to see what their rates are for May but I certainly paid less than 300€/night. Great location, excellent breakfast, lovely staff!
https://www.ducadiyork.com/en/hotel-milan-city-centre/1-0.html
We stayed at the Spice Milano Hotel which is about a block from the Central Station. We paid well under your budget per night for a room with a balcony. Breakfast, if we'd eaten there, would have been extra. There are plenty of places nearby to eat. It's minimalistic, clean and nobody bothers you. While it's not folksy and charming, we stayed at other places like that during our trip.
Hotel Ariston. Believe they’ve undergone recent renovations. Property was a recommend from a flight attendant. Should be within your budget.
Been years but I remember an attentive staff. Had arrived on Christmas day. Took a cab from train station to it. Room was larger than I expected. Good water pressure. Unfortunately I got food poisoning after eating gelato. Sleep, vomiting, sleep, ordered tea. More sleep. Staff checked on me off and on throughout the day. Slept through the New Year’s Eve fireworks. Staff arranged a taxi to Milano Centrale so I could catch my train to Venice.
Traveled in December and enjoyed the Christmas festivities. Booked my Last Supper Ticket with Viator.
( only time I’ve used Viator ) Totally shocked to discover I was the only person visiting the Last Supper. Was just me in the room with the security guard. It was spectacular. Stood there for 30 minutes, staring.
Another vote god Duca di York