New-items play a crucial role in recommender systems (RSs) for delivering fresh and engaging user experiences. However, traditional methods struggle to effectively recommend new-items due to their short exposure time and limited interaction records, especially in dynamic recommender systems (DRSs) where new-items get continuously introduced and users' preferences evolve over time. This leads to significant unfairness towards new-items, which could accumulate over the successive model updates, ultimately compromising the stability of the entire system. Therefore, we propose FairAgent, a reinforcement learning (RL)-based new-item fairness enhancement framework specifically designed for DRSs. It leverages knowledge distillation to extract collaborative signals from traditional models, retaining strong recommendation capabilities for old-items. In addition, FairAgent introduces a novel reward mechanism for recommendation tailored to the characteristics of DRSs, which consists of three components: 1) a new-item exploration reward to promote the exposure of dynamically introduced new-items, 2) a fairness reward to adapt to users' personalized fairness requirements for new-items, and 3) an accuracy reward which leverages users' dynamic feedback to enhance recommendation accuracy. Extensive experiments on three public datasets and backbone models demonstrate the superior performance of FairAgent. The results present that FairAgent can effectively boost new-item exposure, achieve personalized new-item fairness, while maintaining high recommendation accuracy.