I discovered the key to a perfect (and affordable) city stopover

I felt like a competitor in The Amazing Race. “You have five hours in Copenhagen – go!”

My recent trip to Greenland meant a very late arrival into the Danish capital on our return from Nuuk. It was close to midnight when we finally arrived, with departure for Australia the next evening about 7pm.

Given we had to check in at the airport about three hours before the departing flight, we had a window of about five clear hours to explore the city.

It’s always a dilemma. Do you try to see as much as you can, especially if you’ve never been to a destination before? Or do you take it easy and limit your visit to a couple of gems that you can savour more slowly?

It’s a pleasant choice I’m often faced with when on a cruise. I rarely do escorted tours if it’s a city, preferring to research the destination myself and identify some highlights that are off the beaten track – a small art gallery or eccentric museum rather than the popular public gallery, for instance.

Usually, there’s time for a couple of museums and lunch. And, best of all, many of these places have wonderful cafes and restaurants that are often superior to what you find on the street.

In Copenhagen, we checked in to a large, soulless airport hotel. But it was convenient – right inside the airport complex. We’d paid handsomely for a late checkout at 4pm, so we could have a shower before the long flight home.

The receptionist seemed shocked that we didn’t want to pay extra for the exorbitant buffet breakfast. But this was Copenhagen – and there was good coffee and cinnamon buns on offer in the wider city.

I’d been to Copenhagen a couple of times before, but I’d walked everywhere and hadn’t realised how brilliant its public transport system is. The metro line is easily accessed from the airport and our hotel, and it took us to the centre of town in 15 minutes.

It’s fast, clean and simple to use. The line was above ground, so it allowed us some bonus sightseeing of the suburbs. And a lot of gawking at impossibly beautiful, well-dressed people on their way to work.

And I do something smart – I buy and download the all-inclusive Copenhagen Card - Discover, which covers admission to more than 80 attractions and includes public transport, which we could have used to and from the airport if we’d been staying in a city hotel.

It proved to be very useful, even for a few hours. If we’d had a whole day and seen more attractions, it would have been great value, given the strength of the Danish currency compared with the poor Aussie dollar right now.

I bought a 24-hour card on booking.com for a slightly discounted rate of $120 a person. The card comes with 24, 48, 72, 96 and 120-hour options. If we had children in tow, two kids could travel free with each card.

If that seems pricey, consider that a single metro journey costs 24DKK ($6) and a city train pass for 24 to 120 hours is anywhere from $24 to $80. I love the Tivoli Gardens, which is one of the attractions that was free on the card, but it can be up to $50 an adult entry, with only children under three admitted free.

We were interested in the Golden Age of Danish painting, so we visited The National Gallery before the crowds amassed, and the smaller but wonderful Hirschsprung Collection nearby. We strolled in the gardens and stopped for a while at the Natural History Museum, having a cheap lunch in its casual cafe overlooking the park.

Being a hotel tragic, I wanted to check out the posh Hotel d’Angleterre on Kongens Nytorv square, so we took the subway there and sat in the lobby for a while, watching the passing parade. (That bit was free.)

The card gave us two QR codes, one for attractions, the other for transport, and we tapped them happily without problem all day.

If we’d had a few more hours, we could have packed in a lot more.

You can’t get deeply into any destination in that time, but I’m starting to enjoy these fleeting visits and stopovers because there’s no pressure to see it all.

You simply can’t, so you relax and enjoy whatever pleasures you can find in the allotted time.

And that includes cinnamon buns.