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Every (tourist) hostel in the old town of Cartagena in Colombia can arrange a taxi for you at any time. I would think any hotel can as well.
If you're on your own, you could try your best at calling a taxi company and arranging a pickup. But, asking for help at one of the tourist offices would probably be a better bet. Unless your Spanish is good.
If you are comfortable walking through the old town in the middle of the night, taxis are waiting for rides on the eastern corner of the old town, near Monumento India-Catalina.
I've never been to the airport in Cartagena, but considering it's not a small town, I would think that aiming for 4AM would not be a bad idea. But, that's speculation.