24 hours is enough for the essentials. If you're in town for this period of time, make sure every meal is something Balkan - why dilute your experience? I'd do burek and yoghurt for breakfast, ćevapi or a meal at an ašćinica for lunch, and a solidly traditional restaurant for dinner (Nanina Kuhinja, Inat Kuća, Kibe Mahala, or similar). Have a coffee in the afternoon in a very local place (say Miris Dunja Baščaršija or Slatko Ćoše), avoiding the newer trendier locations (such as Ministry of Ćejf or Espresso Lab).

In terms of the segments of the town to visit, I would perhaps avoid the more socialist eastern part of the town (unless that's your thing) and focus on the Austro-Hungarian and Ottoman parts. To do this, stick east of the two UNITIC towers, perhaps even east of the BBI shopping centre.

For specific sites, museums, and experiences, you really should try to go to at least one of Galerija 11/07/95 and the War Childhood Museum. The Sarajevo Tunnel is interesting, but its a bit far out to squeeze it into a day. Skip the National Museum of Bosnia and Herzegovina. Do try and squeeze a tram or a taxi ride in, especially if it's anway taking you somewhere. Definitely walk around Baščaršija and make sure to see the Sebilj, Gazi Husrev-beg mosque, the Latin Bridge, and the Vijećnica. Also make sure to walk along Ferhadija street and see the Eternal Flame. Skip Mt. Trebević, but do walk or take a taxi up to the White Fotress or Yellow Bastion to get a birds eye view of the city.

In the evening, go to Zlatna Ribica for a drink, moving on to Birtija, or Sarajevska Pivara if you have the energy. If you happen to be there on a Monday, you have to go to Kino Bosna. Definitely try local wines (Žilavka or Blatina), beers (Sarajevsko, Gelender, or Brew pub), and rakija (any) in any of these places.