The 21st century challenges, in particular those of climate change, population growth, air pollution and the COVID-19 pandemic global health crisis, demand greater emphasis on infrastructure facilities capable of keeping pace with population needs in well-being, health and economic prosperity. Green infrastructure aimed to intensify ecological processes in built-up areas and deliver vital ecosystem services is of a key significance for Russia, one of the most urbanized countries in the world. The article is dedicated to the issue of providing cities sustainability through creating a basic element of the urban green infrastructure—a public green network—by incorporating and linking a variety of already existing urban environmental components, both spatial and linear, such as natural areas, squares, parks, streets, boulevards, embankments and others. The territory of the case study is Krasnodar region located in the southwestern part of the North Caucasus, the warmest region of Russia characterized by highly varied urban areas. The authors propose the multiple criteria method for evaluating the existing GreenPS and creating a GreenPS framework for six cities located in Krasnodar region both on the southern seaward part and on the northern steppe part. This approach is focused on the sustainable development of the cities, adaptation to climate change and the prevention of local risks with respect to the preservation of ecosystem functions and cultural heritage. The methodology includes three stages: investigating and evaluating the present state of the existing urban GreenPS; defining sustainability potential; and the submission of master plan proposals for the improvement and further development and management of the GreenPS network implementing nature-based solutions. In addition, it follows seven main integral criteria, namely, Accessibility, Density, Sufficiency, Continuity, Diversity, Value and Clean and Sanitary, illustrated by ray diagrams. The results of the study show good potential for construction in Krasnodar region cities of an efficient GreenPS network integrating ecological and social city components. The flexibility of the method proposed makes it replicable for any other city aimed at creating a GreenPS network in sustainable, climate-change-adapted cities.