Here's my post on the same topic awhile back, with replies. I know the dome is closed next week for cleaning and maintenance, but I haven't learned why it can't be visited during the day later this month. It has been open in the evenings but those time slots have been booked.
However, one of those who replied to me, who was in Berlin at the time, wrote: "I was told they were booked for two weeks. However, they did tell me that every day at 12pm they give away 100 tickets for an evening dome climb. They said to be in line at the white pavilion across the street at 11:30am to get a ticket. That was the only way to get in. I hope that helps. PS- I did return to get one of the 100 tickets to see the dome tonight."
I booked a 90-minute tour of the Bundestag for October 18, which ends with a visit to the dome. Unfortunately the tour is in German, so I don't expect to understand much of it, but it will be interesting to see the rest of the building. I also plan to follow our fellow-poster's advice before that and try for one of the tickets they give out at noon. Maybe we'll run into each other there!
EDIT -- As for what's open on Mondays, try the Jewish Museum, DDR Museum, Berliner Dom, a boat ride on the Spree, or any of the open-air sites like Brandenburg Gate, Memorial to Murdered Jews, Tiergarten, Berlin Wall Memorial (but not the visitor center), etc. The Jewish Museum and Berliner Dom require timed entry, the DDR museum sells tickets by the day but not by time.