September does have rain, but it's not as catastrophic as some people make it out to be. The downpours tend to be short and heavy, especially in the afternoons, so the mornings are often clear and perfect for walking around the old town or the beach. What I will warn you about is that the humidity is brutal during that time - you walk five meters and you're already sweating like crazy, so mentally prepare yourself for that.
The good side is real: fewer people, hotels and restaurants with much cheaper prices, and the city feels more authentic without the tourist crush. The risk that nobody mentions is the sargassum thing. That seaweed that sometimes shows up in the Caribbean covers the beaches and smells horrible, and September is when you're more likely to see it. It doesn't always arrive, but it's something worth checking before you go - check Cartagena social media a few days before your trip to see if it's present or not.
If you decide to go, pack a compact umbrella or a poncho, but don't cancel because of the rain. A week in Cartagena during this time is still incredible, just with more patience for the afternoon downpours. That said, if your girlfriend is one of those people who hates getting wet or finds extreme humidity stressful, maybe it's better to wait until October when it starts to improve a bit.