I have a few hours layover in Amman and wondering whether I could use them to see something, rather than stay in the airport. I land in Amman at 4am and my onward flight is at 12:30pm. Assuming that they allow me to get out of the airport (no idea how transit visas work there) and assuming again it takes about 45-60mins to get to the center of the city, I should be able to spend there about 4-5 hours: roughly from 5:30am to 10:30am. So, if this is possible, is there anyone who can recommend a couple of places to visit? Some souqs? A mosque? Is there something like a "historical part of the city"? Some museum worth of an hour or so? I know I get there early, but seeing a city that time it wakes up is somewhat better than reading the newspapers in some airport lounge (and yes, I've had layover in Amman airport before and it's not the most exciting airport in the world).
Also, what options of transport to/from the airport in those early hours?
Any info is very much appreciated.

