Berlin is in the middle of nowhere, there aren't that many preserved towns close to the city... and certainly no Rothenburg. The towns you imagine are all at least 2 hours away by train, in Saxony, Thuringia, the Harz mountains and the Baltic coast... all good places for a few nights away from the big city btw., especially the Baltic coast.
Less than 2h away are Tangermünde, Schwerin and Wittenberg for instance. I would recommend Schwerin, which is the most diverse of the three. There is a nice old town, a beautiful fairytale castle with a park, and lakes (the castle looks best on a boat tour). Tangermünde could be compared to Rothenburg, because of the town wall, but it's a smaller. The town is off the beaten path, but highly recommended. Wittenberg is the most famous one, but also the smallest. And because of the coming celebrations some of the sights could be closed.
If you are one of the persons who would do Rothenburg from Munich (more than 3h by train) you could do Quedlinburg from Berlin. Quedlinburg is just like Rothenburg, but bigger, older, more important, and a World Heritage Site. On weekend there is a direct train between Berlin and the town.
BTW: you could stop somewhere bewteen Munich and Berlin, in Thuringia, Franconia or Saxony, and see dozens of such towns.