Do you use model builder? It takes a few steps but you could buffer your points and rasterize them and use this raster as input as zones for zonal statistics as table. Then this table can be joined to the points based on the ObjectID.
As your points are not in the centre of the raster an extra step is needed to create a point in the middle of the cell to make sure the buffer size extents to the surrounding cells. This can be by converting the points to raster and back to vector again(might be an easier way for this). See example of my model.
Make sure that when you convert the feature to raster you set the original raster to snap raster in the environment settings. You could also do this step by step in Arcmap of course, in that case the snap raster can be set in a tool under environment setting -> Processing Extent.
Hope this works.