I agree, spend all the time in Jerusalem, especially as you won't have any Sabbath constraints.
Obviously you'll want to go to the Old City. The Western Wall is a highlight, as are the Holy Sepulchre and the Temple Mount. (Access to the Temple Mount is sporadic and may not be possible when you are there.) Likewise the bazaars in the Christian Quarter. And for people watching, just about anywhere. For varied types of people, of all religions and cultures: outside the Damascus Gate and around the Jaffa Gate, the Jewish Quarter...
The Mahaneh Yehuda market in west Jerusalem is very photogenic. Mainly fruits and vegetables, lots of local colour both literal and figurative. And if you want to see 19th-century houses, still very much lived in, the little streets to the south of Mahaneh Yehuda, between Agrippas and Bezalel streets, are worth a look.
For wonderful views of the Old City, go to the Mount of Olives. There is also a great view from the Haas Promenade to the south of the city, going towards East Talpiot.
For transport from the airport to Jerusalem, take the Nesher shuttle. It's a 10-seater minibus that will drop you anywhere in Jerusalem. It goes when it's full, and fills up quickly. Follow the signs to "taxis" and you'll see them right there. It costs 58 shekels per person.
When you leave Jerusalem to go back to the airport, you need to reserve Nesher the day before. The phone number is 6257227. Tell them what time you need to be at the airport, and they will tell you what time to be out on the street in any place of your choice in Jerusalem. (If you're leaving from the Old City, where there is no vehicle access, it would need to be somewhere outside, close to one of the gates.)
The company has been going for about 60 years and is totally reliable. If it's a bit late arriving, don't panic. It's making the rounds picking up other passengers. It's by far the best means of transport between Jerusalem and the airport, since there is no direct public transport and you don't want to waste your very limited time getting on and off buses.