As long as you don't have loads of bags and it's not the middle of rush hour, it's perfectly easy to get the train/tram into (and from) town. Take the LRT train from the airport to Zeytinburnu, then change to the tram that will take you all the way to Sultanahmet and beyond. You can also change at Aksaray, but I recommend Zeytinburnu (even though it will take a bit longer) because it's an easier interchange (just over a bridge) and you're getting onto the tram at the start of its journey (so you're more likely to have space to get on), plus there's an Akbil kiosk if you want to buy one of those multiple-trip gadgets. Otherwise, the trip costs 2.20 YTL for two tokens.

Airport, Zeytinburnu.

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