(I'm giving you a nice, late start. Really, there's no reason to wander out early when the weather is crap. Enjoy that warm, toasty bed).
Brunch at Cafe Jasmin, one of Munich's traditional coffee houses (it reminds me of the type of place you would find in Vienna - slightly grandmotherish decor and windows that fog up from the humidity inside and the damp outside). Cafe Jasmin has a few of your typical German breakfast plates - usually combinations of breads, jam, cold cuts, and cheeses. Also, apparently their Milk Rice is the best in town- I haven't tried it yet to confirm that statement, but I trust my friend who passed on that little tip.
(does not take reservations)
Stroll over to the Pinakothek museums (don't forget your umbrella!). Sundays are 1 Euro museum days, and a rainy day is the perfect time to pop into one, or a couple, of the museums. They're all grouped around Königsplatz and slightly to the north of Königsplatz - easy walking distance from Cafe Jasmin. They are all world-caliber art museums with well-curated collections, and a constant rotation of special exhibits. You can't go wrong picking any of them! Some suggestions:
So the Hofbräuhaus isn't usually my top suggestion of a place to hang out in Munich... it's very touristy and there are lots of other great beer halls. However, the Hofbräuhaus on a rainy winter Sunday afternoon is actually kind of wonderful. It's always warm and festive, and there are delicious roast chickens and potato salad. And beer of course. And when you have nothing better to do but hang out with friends in the beer hall, it's just a good place to be. So I would suggest settling in for the afternoon, and if you want to just let afternoon melt into evening...
If, however, you don't feel like parking yourself in the beer hall for hours on end, you could have one beer and a savory snack, take a little evening break, and then do something cultural, like go to the opera. The Bavarian State Opera is a world famous opera house, and located just up the street from the Hofbräuhaus, behind the National Residence. Performances on Sunday evenings usually start early, around 6:00 PM.
Stop in at Tambosi on Odeonsplatz for a late night dinner. They have basic Italian dishes, and if you're not super hungry, they usually have a good selection of cakes, which can be enjoyed with a late night espresso or a glass of wine.
(open until 1:00 AM)