Copenhagen Airport Kastrup (pop 602k) is 6-7km from the city centre.
Lisbon Airport Portela (pop 504k) is also around 6km from the city centre.
And both are capital cities.
Bilbao Airport, Spain (pop 345k) is also only around 4km from the city centre as the crow flies, but due to mountains it's a 12km drive.
Macapá International Airport, Brazil (pop 369k) is less than 2km from the city centre (I've personally walked to this airport)
Val de Cans International Airport (Belém, pop 143k), Brazil is around 4km from the city centre.
Similar, Fortaleza (population 862k) have the airport inside the city, around 2km from the city centre.
And the airport of Leticia, Amazonas (population ~42000) is 200 metre from the city centre. The city is intertwined with Tabatinga in Brazil (population ~59000), who also have it's own airport roughly 800 metres from the city center.
Dushanbe, Tajikistan (pop ~770k) also have a airport in the city. Now I disagree with what Google marks as the city centre, but if we're to use it, then the airport is only 12-1500 metres from the centre.
And staying in the same region, Bishkek's international airport (Kyrgyzstan, pop 900k) is about 2km from the city centre.
Basically anywhere where landmass isn't an issue, it seems airports are very close to the city centre, as building the airports and metropolisation of the cities often happened at the same time.