If "modest budget" is simply "under € 100/night", the B&B Hotel Stazione Centrale (https://www.hotel-bb.com/it/hotel/mi...entral-station) should do the trick. Chain hotel (i.e. a bit soulless), clean, good facilities, good location near central station.

Most of the other good and cheap places, especially in the outskirts, require a bus/tram ride from the stations (or in the outskirts, a car) to get there (e.g. the B&B Hotel Cenisio is usually cheaper than the one at Stazione Centrale, but is located north of the city centre near the main cemetary, in an urban industrial environment close to the tram depot - near a metro station and a direct tram to Duomo though). Avoid the dumps around Lambrate station and Piazzale Loreto.