Just take your (real) passports with you, you need them for identification. Go to Guest Relations on the right hand side of the entrance and say what kind of APs you want. If you are getting the free APs for under 7's, they do this there too. You pay here and get a temporary pass to get you through the gates. Then at some point during that day you need to go to the AP office at the entrance to Discoveryland and they will process your application to give you your real APs, like a little plastic credit card. They will take your photos so all members of your party need to be there and you need to show all their passports. I'm not sure what you mean by what time, but if you mean what is the best time it's hard to predict really. At Guest Relations you may well have a queue, and at the Ap office if they are busy they will give you a ticket and a time to come back I think. We arrived on a Saturday afternoon and Guest Relations were very busy because there were no other ticket windows open by then, but the AP office wasn't busy at all.
PS I looked briefly at the site you mentioned for booking RO and they seemed to do only specific dates, is that right?