Several of those "boroughs" in Munich have parts that are on or next to the main S-Bahn trunk line through Munich, and that would make it most convenient for getting to the airport. It's possible to be in any of those boroughs and have to take secondary transportation (U-Bahn, preferably, or trams/bus)or walk to one of the S-Bahn stations. The S-Bahn trunk runs right through the center of the Altstadt, so there are several stations (Karlstor, Marienplatz, Isartor), not just the Hauptbahnhof, that will get you on an S-Bahn directly to the airport.
I stayed in the Altstadt last year, just a couple of blocks walk south of the pedestrian zone between Karlsplatz and Isartor, and loved the location. It was so close to everything and a short walk to Marienplatz and the Hofbräuhaus. Here is the S-/U-Bahn map of Munich. The MVV website (www.mvv-muenchen.de) has a page where you can put in start at an address and end at Muenchen-Flughafen-Terminal and get directions to the airport.
Not sure what you mean by a B&B. Almost any hotel, except for the high star major hotels, will have breakfast included with your room. If you mean a real B&B (Privatzimmer), I'm not sure how many you will find in the downtown area.
The bus from the Hauptbahnhof makes one stop in Schwabing, near a U-Bahn station, on it's way to the airport, but if you come in on the S-Bahn with a Partner Gesamtnetz Tageskarte, it will cover any other public transportation in Munich that day, or if you use the Tageskarte to get to the airport, it will cover your transportation to the S-Bahn station.