For a taste of post-Soviet Belarus, head north of the centre to the new national library, a ghastly piece of Lukashenko-approved hubris. You'll need your passport to enter the main building, which is in the shape of a giant rhombicuboctahedron (look it up!).
There's a viewing platform and cafe on the 23rd floor; enter around the back. The view from the top is nothing breathtaking, but it's an interesting perspective on Minsk's ever-evolving urban landscape.
The library proper contains a media centre with more than 20,000 vinyl records, art galleries, and a book museum with some surprisingly interesting historical tomes, including a few editions of the Bible printed in Belarusian by Francysk Skaryna in the early 16th century. You can take a guided tour of the library in English for BYN38, or explore it alone for far less.