Saturday, July 2nd at 5:30am, you land in Belgrade, Serbia. You have seven hours.
Here's what you do: exit the aircraft, get through customs, withdraw some serbian dinars and hop on the A1 bus that gets you right into the city center for 300 dinars (approximately $2.70 USD). Note that there are taxi drivers who linger around this bus area in order to offer you a ride into the city for up to five times as much. Don't listen to them. The A1 bus is coming. Get off at Slavija Square (a safer neighborhood than the main railway station in the wee hours of the morning), and walk south to see and say that you saw the Church of Saint Sava in Vračar in person, the largest Orthodox church in the Balkans.
Your next destination is the Belgrade Fortress if you care for a stroll through some really old nature and history at the Kalamegdan park. Walk north from Saint Sava taking Kralja Milana to pass through and take note of some of the city's historic monuments on the way like the Old Palace, Hotel Moskva, the National Museum in Republic Square and then into Kalamegdan. To walk the whole park would take too much out of your limited time so I recommend walking around the park just enough to maybe take note of some of the monuments scattered around the fortress and get a good glimpse of Belgrade's cityscape from the main promenade vista point.
Head back down towards Slavija Square on Kneza Mihaila like a good tourist and explore the shops and cafes in the west corner of Stari Grad. Maybe have a cup of coffee at Koffein off of Cara Lazara, or some ice cream at Moritz Eis, or perhaps a quick bite at the trendy Supermarket Deli off of Topličin venac. Do some people watching, take note of the locals and other tourists alike and maybe make somebody's day by running into them with a familiar face.
When you're ready, take the A1 bus at Slavija Square back to the airport.
Unless of course you get back to the airport to find out that your flight has been pushed back another three hours, your Belgrade experience doesn't end here. Now that you know how easy it is to get into the city, exit the airport once more to explore other areas you couldn't check out in the wee hours of the morning. Take the A1 bus to the main railway station this time (the stop before Slavija Square) and explore Savamala, the urban neighborhood of Belgrade and the up and coming art district... or so it seems. Once you get off the bus and you're walking north through Bristol park into the neighborhood, you'll understand why Slavija Square was a better choice for your early morning, this area was probably kind of sketchy in the morning. Walk past a lot of graffiti and street art and get to Mikser House, a creative hub with locally made designer goods operated out of an abandoned warehouse. Maybe grab for yourself a piece of porcelain jewelry or a graphic notebook as a Serbian souvenir of sort while you're at it. A few more blocks up and there is another design hub you may want to check out called Gradstore - also houses some fine local artists' clothing and jewelry.
By now, you're probably really hot and tired from all the staying up and all the walking around since Belgrade summers are humid and the Savamala roads are awfully hilly. It's time to walk back down to hit the farmer's market, otherwise known as Zelani Venac for your dose of culture before you head on back to the airport. Zdravo your way down the aisles of the market. Maybe make a friend or two. Refresh yourself with a peach or plum. And once you've given yourself enough time to catch your breath, walk down to the main train station and find your beloved A1 bus stop. This station is a bit trickier than Slavija Square since it's the main transportation hub for a whole lot of buses and trains so be sure to give yourself a little extra time to find your bus stop. If you still can't find it, ask the gentlemen in uniforms for the A1.
Congratulations. In seven (or so) hours, you got the feel for a whole city. You started in Vračar, worked your way up to Stari Grad, and had a few extra hours to spend in Savamala within Savski Venac covering all three municipalities that occupy the Belgrade city center. If you have more time, explore Dorćol, Skadarlija, and pretty much the whole northeast end of Stari Grad. That's what I would have done. Also, find yourself some authentic ćevapi or pljeskavica, the traditional Serbian dish one must consume. I'll remember to do that next time. Dovidenja for now!