score:9
Toronto City Airport has regular scheduled flights to Washington DC and New York, and is literally 25 minutes walk from the CN Tower, the unquestionable centre of downtown Toronto. A substantial chunk of that 25 minutes is getting out of the airport. The airport doesn't have a direct transit link to the city.
After that there's a big tie at 13-15 minutes (source: Google Maps, public transit time, directions destination in parentheses).
There are probably many other airports with similar travel times from their city centers; this is a community wiki answer, so other people can expand it with other airports.
Upvote:1
I think Dubai Airport wins this one. It is located in the city itself (technically the area is called Diera); and is only 6 minutes away (3.3 kilometers) to the nearest shopping center / mall.
Dubai Mall (Burj Khalifa) is 14 minutes. Mall of the Emirates (with the indoor ski area) 23 minutes (by taxi).
Dubai itself is a very small place, so nothing is too far from the airport. There is a left luggage facility at Dubai and the metro runs regularly (except on Fridays, where it has a delayed schedule). Taxis are plentiful as well.
Upvote:2
Frankfurt is not too bad. Less than 15 minutes by local train which is very reliable on runs every 10-15 minutes or so.
Downside: it's a huge airport. It's likely going to take a you a lot longer getting from your gate to the train station than the actual train ride. On an international layover you also have to go through immigration and customs on the way out and through security on the way back in.
Upvote:6
The airport in Eilat, Israel is not just close to the city, it's in the city. The terminal entrance is only ~500m from the beach and many hotels: https://goo.gl/maps/Znhsv8FNzBT2
That said, it's primarily a domestic hub, although it does serve some charter flights, and it's scheduled to be replaced next year by a new airport. Visit while you still have a chance!
But as far as major international airports go, it's hard to beat Sydney, which is just 13 minutes from Central station. (To the international terminal, that is, the domestic terminal is just 10 min away!) And yes, Central really is quite central, although you can sit in the train for a few more minutes to get to Circular Quay and see the Opera House if you want.