You might want to spend some time browsing the exhibits at Museum of Astrakhan Culture, Astrakhan City History Museum, or Astrakhan State United Historical and Architectural Museum Reserve. Sights like Museum of Military Glory and Velimir Khlebnikov House Museum highlight some of the local culture. You can watch performances at Astrakhan Theatre For Young Spectators, Astrakhan State Theatre for Young Spectators, and Astrakhan State Drama Theater if you're interested in local theater.