There will be information about this your Navigator (newletter). The reservations can be made at the Excursions desk during a specified interval on the first day. There is no charge, and I did not find it difficult to make a reservation. Both our boys (11 and 8) ended up doing the bridge tour as an Oceaneer Lab activity.
I found it just a little disappointing, because you are not allowed on the bridge itself (sensible), and I've run on the treadmills in the spa, where you have a great view of the bridge equipment. The tour itself admits you to a "secret" deck 9 walkway that goes around above and in front of the bridge, and seems to have no purpose other than hosting bridge tours. Your tour leader is down on the bridge itself and talks to the crowd of about 20-30 up on the walkway. He explains the equipment on the bridge, what it does and how it works, and answers questions about the operation of the ship. It's worthwhile if you're interested, especially if you've never seen the bridge before. But I wouldn't call it a "must do".