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!