The microscopic magnetic and electronic properties of La1-xSrxMnO3 (0.5 <= x <= 0.6) were investigated using Mn-55 nuclear magnetic resonance (NMR). The NMR spectral intensity showed a large decrement as x increases above the boundary (x similar to 0.52) of ferromagnetic (FM) and A-type antiferromagnetic (AFM) phases. The spectrum of the samples shows a coexistence of two different phases, i.e., a metallic major phase and an insulating minor phase. The spin-spin relaxation time strongly depends on the resonance frequency, indicating that the Shul-Nakamura process by spin-wave excitation is the dominant relaxation mechanism. This type of frequency dependence of the spin-spin relaxation time was not observed in the CE-type AFM manganite Nd0.5Sr0.5MnO3. The A-type AFM spin structure and the Shul-Nakamura relaxation process appears to be a unique characteristics of La1-xSrxMnO3 (0.5 <= x <= 0.6) that is related to the 2D-like spin coupling in a nearly tetragonal crystal structure.