We address eco-driving at roundabouts in mixed traffic to enhance traffic flow and traffic efficiency in urban areas. The aim is to proactively optimize speed of automated or non-automated connected vehicles (CVs), ensuring both an efficient approach and smooth entry into roundabouts. We incorporate the traffic situation ahead, i.e. preceding vehicles and waiting queues. Further, we develop two approaches: a rule-based and an Reinforcement Learning (RL) based eco-driving system, with both using the approach link and information from conflicting CVs for speed optimization. A fair comparison of rule-based and RL-based approaches is performed to explore RL as a viable alternative to classical optimization. Results show that both approaches outperform the baseline. Improvements significantly increase with growing traffic volumes, leading to best results on average being obtained at high volumes. Near capacity, performance deteriorates, indicating limited applicability at capacity limits. Examining different CV penetration rates, a decline in performance is observed, but with substantial results still being achieved at lower CV rates. RL agents can discover effective policies for speed optimization in dynamic roundabout settings, but they do not offer a substantial advantage over classical approaches, especially at higher traffic volumes or lower CV penetration rates.