I visited Istanbul on a one day stopover en-route to Kathmandu for my Nepal trip in November 2010. One day doesn’t give you a lot of time to explore, but I found a central hotel (Alzer Hotel), and was able to visit the Blue Mosque, Roman Basilica Cistern, and to potter around the streets taking photographs and geocaching.
Turkish visas are available on immigration - although you have to queue up at the desk beside the immigration officials. A multiple entry tourist visa cost me 15 EUR (cash only) as a British passport holder.
From everything I’ve been told - traffic in Istanbul can be a disaster, but I didn’t have any problems getting from the airport to the hotel in around 25 mins via the coast road via taxi. Official Istanbul taxis are bright yellow and have a Taksi sign on the top. All should have meters - which you should check is running with either “Gunduz” (daytime) or “Gece” (night 12am-6am and 50% more expensive).
After visiting the Basilica, I walked round the corner away from the really busy restaurants looking out onto the square, and found the House of Medusa hidden behind a quiet courtyard (my review) for a great Turkish kebab.