48 Hours in Istanbul

The New City (outside the tourist quarter) is best for your evening arrival - more lively and tons of foot traffic and you can see Galata Tower (at least from the outside unless the viewing area on top is open). I would overnight there (Ikstiklal Street, Beyoglu, etc). On the other hand, Sultanahmet/Old City would be best for seeing the major attractions since they are in a compact area (you can take the tram over). That area would be dead at night though, which is why I wouldn't recommend staying there even though closer to most things tourists want to see. You'll have to be brutal in prioritizing what you want to see given the incredibly short time you have, so read up on each attraction and decide. Definitely include at least one mosque and a museum (my favorite was Topkapi Palace but I spent literally all day there). I haven't been to the new airport - maybe it's best to play by ear and see how long it takes you to get in before deciding on how much time to give yourself to get back.

Edit: Messed up the timing. We actually arrive at 4:00 PM and fly out at 7:00. Gives us a bit more time.

Because they're using the new airport, no one really knows how long it will take to get to/from the city. I'd stay in/near the Sultanahmet quarter - where the sights are. There are plenty of restaurants. I ate at Olive Anatolian Restaurant last June. Excellent food, reasonably priced, wonderful views. Make a reservation and ask for a table with a view. As you can see from the photos, you'll see the views easily, but it's better if you're sitting at a table by a window. I'd just walk around on first evening. Istanbul's very pretty after dark and quite safe (watch out for petty theft).

Rustam Pasha is around the corner from the Spice Bazaarand no lines. Hagia Sophia and Blue Mosque are near each other, can be longish lines at both. From there you can hop on the tram to the Grand Bazaar. Tram are usually the fastest way to get around - taxis get bogged down in the slow-moving traffic and walking can be faster! Many taxi drivers are dishonest (though certainly not all). Agree a price before you get in and pay attention when you pay. Make sure the notes are intact and note the denominations. If you only have large notes, say to the driver - I'm giving you a 50 lira note.

Sights close on different days. Eliminate those which are closed on your day. Mosques close for prayers. They shoo out people some 15-20 minutes before prayer times and reopen after worshippers have left. The ones likely to affect you are at mid-day and late afternoon. Use google to find the times for the day you are there; times are determined by the position of the sun, so they change daily.

The new airport makes it unclear about how many hours to pickup luggage, clear passport, clear customs, find a taxi or bus, etc. etc. and then travel a long distance into Istanbul.
Stay in Sultanahmet. Choose a hotel/guesthouse within 10 minute walk of Hagia Sophia, there are many, that suit your needs. Since your time is so valuable, arrange with them, for a set fee, for transportation from and to airport. Maybe negotiate a package price that includes transportation, bed, and breakfast.
You will arrive either completely exhausted or exhilarated, so you then choose, bed or walk. That area is beautiful in the evening. There is a large park with fountain located between Blue Mosque and Hagia Sophia. There will be many local people out for a stroll and an ice cream or hot corn on the cob etc.
Full day spend at Blue Mosque [free, it is a functioning mosque, place of worship; they will provide bag for shoes and scarf to cover hair] Hippodrome, then Hagia Sophia [an hour or more] and Topkapi [three or more hours]. Underground Cistern is wonderful if you have a spare 30-45 minutes. Return to hotel for rest if needed. Tram to Grand Bazaar, or walk [it is about 30-45 minutes, more with enjoying the street, getting some baclava or ice cream] and an hour or two wandering in the Bazaar. Be careful with all of these locations about open days and hours.
Partial day, leave luggage in hotel, take tram to Kabatas [IIRC 45 minutes to one hour and visually enjoyable] and take the funicular up to Taksim Square. Walk along Istiklal [pedestrian only, visually enjoyable] for the couple Km [1-2 hours] to where it is close to Galata Tower. The lines might be too long to wait, but maybe you will be lucky. Then the Tunel Funicular down to the waterfront where you will get the tram back to Sultanamet [20-30 minutes] and pick up you luggage and leave for the airport. These times can lengthen during rush hour, or accidents etc. etc. Don't try to squeeze the time to get to the airport and through security screening to enter before you get to check in [if it is the same as the previous airport] and then security of person and carry on, and then passport control [Turkey examines passports coming in AND going out] and then waiting for your flight.
Have a wonderful time, and start planning on your return!
Harvey

You can do the Hippodrome, Blue Mosque, Cistern and Hagia Sophia in one day. Find a hotel in the old district walking distance from these places. The Grand Bazaar is interesting, but we were not interested in shopping, so we didn't spend much time there.

The Archaeological Museum is amazing, highly recommend that as well at Topkapi Palace. The Galata Tower is a bit out of the way from the old city.