Another vote for the small towns along the Harz as mentioned in the first post!!! They have UNESCO World Heritage status and there is reason why they obtained it.

These places can easily be reached by train, there is absolutely no need to bother with a car. Connections on regional trains are frequent and easy.
You may also want to consider the narrow gauge steam train from Wernigerode up to the summit of Brocken.

Forgive my blunt German honesty, but focusing on nothing but big cities is a common mistake among visitors from overseas. It is the small places where you find authentic old town centres that survived World War II relatively unharmed. If you want to get a feel of old Germany, visit some small towns. The Harz region can easily fill your 4.5 days.

I spent the first 22 years of my life in that region, so if you want more details, you are welcome to ask...