I studied in - I studied at

Use - At - mainly to indicate time or place  >>> I will be at home at 5pm 

Use - IN - mainly to indicate you are inside a place >>> the keys are in my pocket, I will be in Paris on Tuesday

But there are numerous other instances when these two words are used like>>> in order to, to be out in the rain, the train will arrive in a minute or so. To be at rest,  I saved and I paid my debt at last  etc etc

 "I studied at Cartagena University" is correct.

You can say, however, "I studied in Cartagena" as in, in the city of Cartagena in general. I'm not sure why it's okay that way and not for the university, but that's how I'd say it.