I am arriving in Istanbul from the US in April at 3:45 pm. I want to fly on to Goreme but have not bought my plane ticket yet. There are several options to ASR (Kayseri) but I am not sure how much time I need to allow to get to the domestic gate. Could I make the 6:15 flight? If that is possible I would not reach ASR until almost 10 pm. Could I then get a shuttle to my hotel in Goreme that late at night? Any advice?
I think that 2 1/2 hours is enough to make the ASR flight. Are both flights on one ticket? I notice that Turkish Airlines has the flight time of IST > ASR at 1 hour 25 minutes. That would put your arrival at 5:40, not 10 pm. If you have questions about transportation from the airport to your hotel, I would contact the hotel.
IST is indeed a very big airport, and it's not uncommon to have surprisingly long walks to get between points (which is not necessarily a bad thing, if you've just come off of a very long flight - I appreciated the chance to stretch my legs). If everything goes well, you should be fine, though I wouldn't say your layover was excessive.
The real question is, what if everything doesn't go well? If both flights are all on one ticket, and you miss your connection (because your inbound flight arrives late), then Turkish Airlines (TK) will get you to ASR on the next available flight, so that reduces your risk. Does TK have later flights for IST to ASR that you could catch later on the same day? If yes, you'll be fine; if not, you might be looking at an unplanned overnight in Istanbul. If both flights are not on a single ticket, then you might have to pay to buy a new ticket to ASR if you miss your connection (domestic flights in Turkey are not expensive, so this would not be a financial disaster but it's still an extra expense, plus a possible night in a hotel).
If it's all on one ticket, and if TK has later flights that same day you could take if needed, then I would go for it.
Check with your hotel in Goreme about airport transport logistics at your potential arrival time(s). They will know better than anyone here and can hook you up with what you need.
I will share my recent experience arriving in Istanbul and later flying to Cappadocia and arriving in Goreme.
Recently we flew into Istanbul's big new airport to the northwest from Sicily. We arrived about 11 pm, so we stayed in a hotel actually in the airport for about $200. This was expensive but wise. The next day, we found that there were about a dozen buses to Istanbul about 40 minutes. Upon arrival, we had walk two blocks to a tram station, figure out how to buy a tram ticket which was surprisingly difficult - made harder by scammer constantly trying to "help" me for money. Then take a tram into the city center.
A few days later, we flew to Cappadocia. However, we flew to Nevsehir and took a shuttle 30 minutes to Goreme. The flight was on Turkish airlines and was only $30 each. However, it was from another airport to the south east of the city. We paid about $30 to take 3 people the hour drive. Then took the flight (2 hours I think). There were tons of taxis wanting a lot of money, but we found a big shuttle bus which was pretty reasonable. On the way, I got onto an app, and booked a hotel for our group, then told the driver, and they dropped us off at the actual hotel in Goreme - which was an interesting, unique, and a bit of strange place.