It is just this district, located under the Kremlin`s hill, that has the atmosphere of an old Russian town. The district was so called because of the river Pochayna, as it is located across the river. Hence the word Zapochainie. It has original old buildings of past centuries, a lot of churches. Their onion-domes create very colourful picture of the old Russia.
If you simply want to have a promenade, you should go to the Upper Volga River Embankment in the evening. It starts at the walls of the Kremlin, near the monument to the pilot Chkalov. At night there are always a lot of walkers almost in any weather. They saunter, communicate, and make friends. Besides the Embankment, there are a lot of promenades, starting at the Kanavinsky Bridge and going towards the Kremlin.