Contains a surprisingly good (for a town that is not even a regional center) collection of Russian art of veginning of 20th century, that includes some names like Feshin that are not easily found for instance in Moscow. The story behind is that in 1918 an exhibition of Kazan artist was travelling on a boat on Volga and was stuck in Koz'modem'yansk during the Civil war. The local painter Alexander Grigor'ev founded the museum and made sure that the exhibition went there rather than was destroyed or plundered. In addition, a nice historic museum. The building, Ponomarev House, is a merchant stone house from 1883.
Actually, not all the houses are wooden, there are some nice ones made of stone. Two examples from 19th century are in the photos.
Basically, the whole historic part of old Koz'modem'yansk (about a kilometer size) is a collection of wooden houses with beautiful wood carvings. The addresses are in the photo captions.
From Cheboksary (bus station next to the train station, cross the reilway tracks in a tunnel) take a bus, 1h30. The most practical way (and, actually, the only way) to see Koz'modem'yansk in one day from Moscow is to take an overnight train to Cheboksary and a subsequent bus to Koz'modem'yansk. It arrives around 11:00. From Koz'modem'yansk bus station, take a minibus to the historic part. Then you have about 4 hours. Take a bus back and an overnight train to Cheboksary. The train schedule is here .