Don't even think about taking any mode of transport other than the trains- with the bus, any congestion and your time in Tokyo proper will be ruined.

Given your limited time, I'd take the Keisei. Take the Access Express (not the Skyliner, which goes to Nippori & Ueno) toward Haneda Airport. It stops at Higashi-Ginza which puts you near the action in one hour flat. Fare is around 1300y per person, one-way.

Keep in mind that the Narita Express is considerably more expensive than a comparable Keisei fare, which is usually faster as well.

As I see it, you'd have 2 or so hours at most. If your inbound flight is on-time/early and immigration lines are not too long, being in First you have a chance of catching the 10:06 train, which puts you in Ginza by 11. Start heading back around 1:30 or so to be sitting in the lounge at 3pm. AA lounge is really nothing special, I'd spend the 45m you have in the JAL lounge instead.

With that little time, do some research about one or two things you want to see or do and go straight there. Skytree wouldn't be a bad idea- it's a new landmark and there's a shopping mall at its base which you could buy stuff and eat at. I think the Access Express stops at Oshiage which is right next to Skytree too.