Urban flooding is intensifying with urbanization, climate change and ageing drainage infrastructure, particularly in data-scarce foothill cities. This critical narrative review, based on a transparently reported selection procedure informed by PRISMA 2020, examines how geographic information systems (GIS), hydrological–hydraulic models (EPA SWMM, StormCAD, HEC-RAS and related 1D/2D tools), green infrastructure (GI/LID/SUDS) and machine learning are combined in urban flood management, and what this implies for Central Asian cities drained by open aryk networks. Searches of Scopus, Web of Science and Google Scholar, supplemented by Russian and Kazakh-language sources and citation searching, were used to select 33 sources published in 2002–2026 (28 from database searching and 5 from other methods), of which 26 are primary case or modelling studies. A structured data-extraction table summarizes the study area, tools, input data, calibration/validation and quantitative outcomes of each primary study. GIS–hydrological–hydraulic model integration was the largest thematic cluster (14 of 33 sources), whereas calibration or validation against observed flood data was reported in only some primary studies and several reported none. Reported GI/LID effects are not directly comparable because they refer to different metrics and design storms: LID covering 9.81% of a catchment in Palembang reduced runoff volume by 21.0–35.4% and peak discharge by 11.3–37.7%, whereas a combined GI scenario in Istanbul reduced the flooded area by 36.94%. Peer-reviewed evidence on urban drainage in Central Asian cities remains scarce. We recommend combining digitized aryk networks, high-resolution DEM/LiDAR data and locally derived design rainfall with calibrated 1D/2D models and GI scenarios, and we outline research priorities for the region.