Hiroshima Airport is a 45-minute to 1-hour bus ride to town. There is a Limousine Bus Service from the Airport to Hiroshima Shinkansen Station (45 min), Nakasuji station on the Astramline (40 min) or Hiroshima Bus Centre (50 min). If you want to be at the town area, take to Hiroshima Bus Centre. The above website is one that even shows the map of the route, so if you are living in a hotel/hostel, check for the nearest stop.
This bus is 1300yen (about S$20) for 1 way, and is slightly cheaper if you are going 2-way. The bright red pillar shown in the picture is the bus pillar for the one that goes to Nakasuji station and Hiroshima Bus Centre. Japanese are normally rather disciplined, and without a line drawn on the floor, will still queue up in a straight line after purchasing the bus ticket. Note: you have to purchase the ticket in the building before queuing or getting on. Upon getting on, the driver will collect your ticket from you.