I have a raster index map and and also a polygon fishnet. I would like to assign mean index value into the grid cells.
How can I do that in ArcMap?
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If by "grid cells" you mean the fishnet polygons, then use the Zonal Statistics as Table with the following parameters:
Input raster or feature zone data = your fishnet Zone field = a unique ID field in the fishnet Input value raster = your ECI index raster Output table = output table Ignore NoData = ticked Statistics type = MEAN
Then use the zone field to join the output table back to the fishnet using either the Add Join tool or the Join Fields tool.