There are direcrt flights but two days in each will mean you really not see either, especially if it includes travel time.

Esbelli Evi is highly recommended.

It might be too early for the flights... If you only have two days in each, you can make it work.

We had three full days in Cappadocia, but the distances are generally small and if you don't mind a bit of a whirlwind tour, you can made it worth your while. You will need to rent a car to take advantage of your time. We loved the Kelebek Hotel in Goreme.

For example, in a single day, we did one of the underground cities (Derenkuyu), climbed Uchisar castle, walked one of the valleys, spent a couple of hours at Zelve open air museum, and wandered Goreme before and after dinner. It was a busy day but didn't feel overly rushed. On another, we went ballooning, had a two hour breakfast in one of the valleys (provided by our hotel), drove to another of the valleys, saw an old roman bath, and old cave monestary, had a great dinner with a group of fellow travelers, and went to the Deverish show.

As for Istanbul, most of the historic sites are quite close to one another. You can definitely the Blue Mosque, Hagia Sofya, the archeological museum, and the spice market in the same day. Then do Topkapi Palace, the Grand Bazaar, maybe the Basilica Cistern, and the Galata bridge on day two.

You'll have a lot to pack in, but don't let that dissuade you.

The schedules are up, I just checked. Pegasus does not fly from Istanbul's Ataturk airport (IST) everyday. They do fly from Istanbul's Sabiha Gokcen (SAW) airport (on the Asian side) everyday. I was in Turkey in October. I flew Turkish Airlines from IST to Nevsehir (NEV). Good information can be found at http://www.goreme.com/flight-tickets.php. I flew from Kayseri (ASR) on Pegasus as I was taking Pegasus via SAW to Rome.

I, too, can recommend the Kelebek Hotel in Goreme. I loved the location, the staff and the atmosphere of the hotel.

If you stay in the Sultanahmet or Sirkeci area of Istanbul it is a bit more difficult to get to SAW. However, they are ideal areas to base yourself in Istanbul. Walking distance to all the historical sites.

I have flown out of both Istanbul airports. Gokcen is fine as is Pegasus. I, howeve, prefer Turkish Air if available.

Are you checking Nevesehir airport also? It seemed just as close if not closer than Kayseri.

I think you're better off choosing one or the other depending on your interests. There's plenty to do in Istanbul to fill your four days including a day on the Bosporus if the weather's nice. And four days in Cappadocia would give you a chance to explore the area at a more leisurely pace, and perhaps go for a hot-air balloon ride. We stayed at the Gamirasu Cave Hotel when we visited Cappadocia, and really enjoyed it. It is situated in a small inhabited village. The staff were friendly and helpful, the rooms comfortable, and the food delicious. They are also able to arrange transportation and / or day tours in the area.

you can also check for turkish airlines flights direct to Cappadocia (NAV) airport.
3 nights is perfect to discover around.
we stayed in Uchisar which was not crowded with tourists and our hotel Taskonaklar had wonderful view over Cappadocia with really comfortable rooms and a large breakfast

I must admit that I would find it very frustrating to try to fit both Istanbul and Cappadocia into 4 days. I spent 5 full days in Istanbul and 3+ in Cappadocia, packed a lot into each day, and still feel like a skipped a great many things that I would have enjoyed. But to each his/her own!