The Pinacoteca di Brera will be open from 8.30 (unless you arrive on a Monday when it’s closed)
For breakfast you can get a cappuccino and pastry in any of the bars around here, but nothing more then that really. You’ll notice that most people order coffee at the bar, drink up and leave, breakfast is not an important meal here.
There are several places in the Via Pontaccio area for having breakfast. There is a Princi bakery at the beginning of Via Mercato . Also a very nice place called Viole di Liquirizia in Via Madonnina, in the heart of Brera. The latter is a very small place so you won't be able to lounge there for long while at Princi it will be easier to camp out for a bit while you wait for your check in time at the flat.
"nice place to have coffee/ breakfast?" - The quintessential "quaint and charming lane" of Brera (and quite a magnet for tourists) with plenty of cafe-bars and restaurants is Via Fiori Chiari running parallel south of Via Pontaccio. You could explore around here and take your pick; or avoid it and try elsewhere, up to you.
That Princi bakery is actually part of a chain with some repute. The one that Aroboro mentions I think is actually their "Via Ponte Vetero, 10" branch, which is a continuation of Via Mercato, south of Via Pontaccio, towards the intersection with Via Cusani and Via dell'Orso. You can see it here [ https://goo.gl/maps/gMdPGfQS4H92 ]. A 600 yard/meter walk north of Via Pontaccio towards the Moscova Metro Station, will get you to the nice piazza on Largo la Foppa where there is another Princi branch in an attractive spot to sit down and watch Milan rush by, which you can see here [ https://goo.gl/maps/Xdiy4Ejua792 ]. (For your information if you don't already know: you can click and drag around the Google Streetview to explore the location. To see a regular Google Map of the location, just click on the left arrow inside the black box near the top left of the page. Princi website in English here [ http://www.princi.com/locations/ ], and in Italian here [ www.princi.it/ ], and of course you can find each of their branches on Tripadvisor.)
For museum opening times, you can check many of them here on the official municipal government website [ https:/…Musei_Civici_Milano ] - many are open by 9 am. To the west, you could also go for a walk in the large and green Parco Sempione surrounding the historical Sforza Castle. The museums of the Sforza Castle will also be open by 9 am.
An alternative to the east, if you walk to the other end of Via Pontaccio which becomes Via Fatebenefratelli, it will end at Piazza Cavour. Piazza Cavour is where you will find the medieval Porta Nuova arching over Via Manzoni - I believe it is one of only two medieval city gates (the other being the medieval version of Porta Ticinese) left standing along the old circle of canals (Cerchia dei Navigli) now all paved over as a ring road. (The Napoleonic version of Porta Nuova, built along the line of the historical Spanish Habsburg walls of Milan, can be found if you turn north [left] a little earlier along Via Fatebenfratelli, and head straight to the end of that Corso Porta Nuova - relatively far way, and no longer in a straight line from its medieval progenitor.)
At Piazza Cavour, you have the choice of taking a walk in the Giardini Pubblici at the north end of the piazza; or take a detour down the narrow Via della Spiga (parallel to the larger Via Senato which Via Fatebenefratelli changes into) that forms the northeast side of Milan's world famous Quadrilatero della Moda (Fashion Quarter), and take a look at the shop windows that you pass by. You will find the women's fashion flagship stores of Dolce & Gabanna as well as Moschino on this street. I have an interactive Google map of Milan that marks with orange pins, over ten flagship stores of the famous fashion designer labels that have made their headquarters in Milan, there in the Quadrilatero della Moda [ https://goo.gl/0UUGNP ] - just zoom-in not far northeast from the heart of Milan, the Duomo.
If you wanted to complete the Quadrilatero, you would turn right at the end of Via della Spiga into Corso Venezia, When Corso Venezia ends at Piazza San Babila, turn right again and look for Via Monte Napoleone (the most famous street of the Quadrilatero); turn right into Via Monte Napoleone, and then at its end, turn right again into Via Manzoni to complete the Quadrilatero. All of that, and the stopping and looking, will probably take more time than you have, so just calculate how much time you need to get back to where you need to be, and start heading back when that time comes.