Weekend in Kazakhstan means monuments-meals-public places. We aren't sure if you'd like to see/try something else. Most of the food places in Astana (currently Nur Sulton) or Almaty expect guests even from Friday's night and if you want to try something special - please, be fast about it!
We suggest tours to these cities by train or tour to Nur Sulton done by local avialines - that's how you can save your time. Some people combine weekend tour with Kazakh holidays, so they can get invited to the common people's chill and rest by sudden.