Fly from New York LGA or New York JFK to Cartagena, Colombia for $322 round-trip on Delta Airlines. 

Travel days will vary per month, but you will have the most luck finding seats departing on Tuesday, Wednesday, and Saturdays. The travel window for this fare is from late August through Delta's end-of-schedule in March 2019.

Searching on Delta's website, I could find seats departing LGA on Wednesday, September 19, and returning from CTG on Saturday, September 29, 2018.

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If you need a place to stay while you're visiting Cartagena, here's what's available for the listed dates on Hotelwatchdog.