Prizes are always triggered by bust-outs, and random prizes aren't triggered by any specific bust-out or any specific time. So, to have the random prizes awarded at a specific time or specific bust-out, you'll probably need to create the prizes with the Recipient set to "Manual", and then use the "Random Player" button on the Players tab to randomly choose a player. You'll then need to edit the prize and select the player in the Override section to award it to the randomly chosen player.
To announce the prize you'll have to display an ad-hoc message. From the Controls tab press the "Display Message" button. Problem here is that, since the message wasn't triggered by a specific event, there's no contextual information, so you'll have to enter the message completely by hand - meaning you can't put something in like <name> to display the recipient's name, or <winnings> to display how much the prize is worth, etc. You'll just have to enter the message exactly as you want it to appear.