Know Before You Go
The easiest way to get to the museum is to take the tram number three (3) from the Old City/Baščaršija away from the direction of the Old City. Get off when you see the American Embassy on the right (it's very large and a commonly referenced landmark in Sarajevo) or the Museum itself on the left. You can purchase tickets for around €1 at most kiosks.
If you would prefer to walk, which is recommended, go from the Old City to the river and head to the right, following the river along the same street for around 20 to 30 minutes. At some point the street will pull away from the river but will reconnect in about 300 meters and continue until the National Museum of Bosnia and Herzegovina which will be on you right.