In all my travels Cartagena has to be the most difficult placesI have ever encountered for posting a humble postcard. There's no real post offices so to speak of and the ones that do exist are hard to find and have the most inconsistent and inconvenient opening hours.
So the best option for most travellers (in my opinion) is to go to this SPECIFIC souvenir shop located on the corner of Calle Roman and Calle el Cochero del Gobernador. It's right near the Plaza of the Proclamacion and the Cathedral.
Here you can:
1. buy postcards
2. buy international stamp for postcards
3. post your postcard in a 4/72 mailbox
Will it be slow? Hell yes.
Will it arrive? Hmm.. I'd say 6/10 likelihood
Is it cheap? International postage stamp is 10,000 pesos and this souvenir shop charges an extra 2,000 pesos to buy it through them.
Is the souvenir shop listed on any map making it easy to share? No. But honestly it is easy to find. Follow Calle Roman until it intersects with Calle Cochera del Gobernador. Or even easier, go to Plaza de la Proclamacion and look for the nearest souvenir shop.
Is it worth it? That's for you to be the judge. For some it will be a cute act of faith and the fact you can do all 3 steps in the one location that is easy to find with convenient opening hours is worth the 2,000 pesos markup.
After affixing your newly purchased international stamp you can insert your postcard in the blue post box located at the entry to the souvenir shop and is labelled 4/72 (apparently the name of the apparently non-existant Cartagena/Colombian postal service)
Wishing you the best of luck! And long live snail mail and cute traditions!