For a taste of post-Soviet Belarus, head north of the centre to the new national library, a ghastly piece of Lukashenko-approved hubris. The building is a giant rhombicuboctahedron (look it up!) that is lit at night and contains more than two million records as well as art galleries and a viewing platform on the 23rd floor.
You can take a guided tour of the library in English for BR100,000, or explore it alone for far less: just go to the registration desk in the lobby with your passport. The view from the top is nothing breathtaking, but it's an interesting perspective on Minsk's ever-evolving urban landscape.