This questiom states that the distance was 400038km with a lunar altitude of 254 km. As gateway will have a max. 70,000 km distance from the lunar south pole, it may break the record. However, because it is in a NRHO, and there will only be astronauts onboard for short periods of time, the maximum distance may be less. Can anyone find an answer?
Whether or not this does happen, that's a trivial and fleeting milestone. Apollo 13 was not known for that trivial milestone. It was known for being a massive failure that was turned into a non-failure.
The answer to your question will depend on when the astronauts launch and return. The will spend about 5 days traveling to the Moon, 5 days traveling from the Moon, and 11 days in NRHO (or transit from / to the Moon). That's about a third of a month. The launch date is not known. If that third of a month happens to occur around lunar perigee, they might or might not break the Apollo 13 record.
The intent of Artemis II is not to break a trivial Apollo 13 milestone. It is to show that Artemis III will likely be a viable mission. In turn, the goal of Artemis III is not to set the distance record that humans traveled from the Earth. The goal is to spend more time on the Moon than did any previous mission to the Moon. Missions beyond Artemis III will spend even more time on the Moon, with the goal of eventually having a permanent presence, with a long-term goal of learning how to go to Mars.
That trivial Apollo 13 milestone will of course be broken by a trip to Mars (and back). The more important lesson-learned from Apollo 13 was how to recover from a very serious, life-threatening failure, and that will live on. There might well be serious failures on the journey to Mars, stay on Mars, or return from Mars.