While it has been over five years since I have been to Cartagena, my wife and I love it.
First the flying bit. One runway coastal airport, last I was there it only had a VOR approach, no radar, with higher minimums that normal, reason; a hill just to the west of the runway. There are two parking areas, one on the main terminal ramp, where you will most likely be parked unless the airport is overly busy, the other is on the other side of the runway directly opposite of the main ramp. You and your passengers will have to go through customs, both inbound and outbound. But it is quick and easy, never had any problems.
ATC is simple and straight forward and in my experience, the local controllers speak better English than many in Bogotá. As for fueling, like nearly all airports in that part of the world, airlines come first. However, there is not all that much airline traffic and I really never had a problem. Catering from the airport is so, so. If you need really nice catering, I would recommend using your hotel.
I used Universal's handlers down there, I cannot recall the name of the company now, but one of the employees is Silvia Padilla (May have spelled her last name wrong), if she is still there, please tell her that Alec and Suzi send her our regards and that we hope all is well. If she is still working there, she will take very good care of you and your passengers.
If possible try to stay in the old city/fort area. The best hotel is the Santa Clara, a true five star hotel, a bit expensive but worth it. Next recommendation is the Hotel Charleston, another really nice hotel. We've stayed in both, they are in the old fort area. In the old fort area there are small squares all over the area with restaurants and bars everywhere. These restaurants nearly all have really good food and reasonably priced. You can buy beer from little sidewalk stores, really cheap. With these restaurants in the squares you will have a choice of eating inside or eating in dinning areas in the square itself that each restaurants lease. I would highly recommend eating outside in the square.
As for hotels outside of the fort area, we've stayed at the Hilton Cartagena, which is a very nice hotel, another place that I would recommend. Since we have been there, I know that there has been at least three more four star hotels built in Cartagena, however, I don't know anything about them.
Overall Cartagena is a very safe city, especially in the old fort area. We've never had a problem in all the times we have been there. Also back then there was a large presents of US Military personnel, all wearing civilian clothes, you can tell them by the real short haircuts.
Have a great time, wish I was going with you.