With the rapid advancement of cloud-native computing, securing cloud environments has become an important task. Log-based Anomaly Detection (LAD) is the most representative technique used in different systems for attack detection and safety guarantee, where multiple LAD methods and relevant datasets have been proposed. However, even though some of these datasets are specifically prepared for cloud systems, they only cover limited cloud behaviors and lack information from a whole-system perspective. Another critical issue to consider is normality shift, which implies that the test distribution could differ from the training distribution and highly affect the performance of LAD. Unfortunately, existing works only focus on simple shift types such as chronological changes, while other cloud-specific shift types are ignored. Therefore, a dataset that captures diverse cloud system behaviors and various types of normality shifts is essential. To fill this gap, we construct a dataset CAShift to evaluate the performance of LAD in cloud, which considers different roles of software in cloud systems, supports three real-world normality shift types and features 20 different attack scenarios in various cloud system components. Based on CAShift, we evaluate the effectiveness of existing LAD methods in normality shift scenarios. Additionally, to explore the feasibility of shift adaptation, we further investigate three continuous learning approaches to mitigate the impact of distribution shift. Results demonstrated that 1) all LAD methods suffer from normality shift where the performance drops up to 34%, and 2) existing continuous learning methods are promising to address shift drawbacks, but the configurations highly affect the shift adaptation. Based on our findings, we offer valuable implications for future research in designing more robust LAD models and methods for LAD shift adaptation.