Effective management and prioritization of Architecture Technical Debt (ATD) is essential for modern software development. This systematic mapping study provides a comprehensive overview of the methods and approaches employed in these areas from 2018 to 2023. The accumulation of ATD in a software architecture can lead to increased costs, decreased maintainability, and reduced system quality over time. This systematic mapping study aims to assist researchers, practitioners, and software development teams in understanding the challenges and solutions associated with managing and prioritizing ATD. This systematic mapping study concludes with insights that contribute to more informed decision-making related to ATD by analyzing the current research and available knowledge. In this systematic mapping study, relevant studies from four academic databases are collected and categorized. We started with a total of 75 studies based on our initial criteria, later simplifying the research to 40 ATD management studies. The various ATD studies included tools, models, metrics, and microservice architecture related to ATD management methods. The analysis focuses on identifying key themes, research trends, common practices, and future direction related to these categories of ATD management. We discovered that the creation and optimization of ATD management tools and models is a sure path to accurate risk assessment and refactoring automation. An additional analysis was performed to identify ATD prioritization methods based on management metrics such as criticality, refactoring cost, and architecture smells. Architecture smells and their prioritization have also been heavily researched, and we found that they will continue to be because of their architectural significance and pressing prioritization opportunity. ATD management and prioritization is critical for the success of long term software system maintenance.