When searching for “the spirit of Olympic Sarajevo”, it's best to start at the Olympic Museum of BiH. The old museum was engulfed in flames in 1992, so the collection is now located in the same building as the Olympic Committee of BiH, next to the Juan Antonio Samaranch Olympic Stadium (Zetra).
You should also definitely make a point of visiting the Olympic Mountains – Trebević, Bjelašnica, Jahorina and Igman – and it would be interesting to visit the Sarajevo neighborhoods and former Olympic Villages, Mojmilo and Dobrinja, which were built as part of the preparation for the XIV Winter Olympic Games.