Heat stress (HS) and water stress (WS) pose severe threats to viticulture, and effective management solutions to counter their effects on grapevine performance must be examined. In this study, we evaluated the physiological and spectral responses of Vitis vinifera L. cv. Sauvignon blanc to individual (HS) and combined (HS + WS) stress under four different cooling and irrigation strategies. The treatments were: standard drip irrigation (SI), extra drip irrigation (SI+), extra sprinklers irrigation (SPRI), and sustained deficit irrigation (SDI; 50% of SI). Compared to the other treatments, in the early stages after the occurrence of HS, the vine water status of SPRI and SI+ improved, with high stomatal conductance (gs) (SPRI) and stem water potential (Ψstem; SPRI and SI+). All the physiological indicators measured were significantly lower after the end of HS in the SDI treatment. We also identified the spectral response of grapevine to HS and combined HS and WS (resulting from SDI). Consistent with the physiological analysis, the proximal spectral responses of leaves identified SPRI and SI+ as putative cooling strategies to minimize vine HS. The vines undergoing combined stress (SDI) showed greenness amelioration 10 days after stress, as revealed by the greenness vegetation indices (VIs), i.e., Green Index (GI), Normalized Difference Greenness Vegetation Index (NDGI), and Visible Atmospherically Resistant Index (VARI). However, their physiological recovery was not achieved within this time, as shown by the Simple Ratio Index (SRI), Transformed Chlorophyll Absorption Ratio Index (TCARI), and TCARI/Optimized Soil-Adjusted Vegetation Index (TCARI/OSAVI). A three-step band selection process allowed the identification of the spectral traits’ responsive to HS and combined stress, i.e., 1336–1340 nm, 1967–1971 nm, and 600–604 nm.