There is two ways of doing this, the best is with Spatial Analyst using Extract by Attributes using the query value < 1.5, this will put NODATA in the areas where the height is greater than 1.5m. If you want to cap the raster to 1.5 so that the values are 1.5 where the raster exceeds 1.5 then use con:
Con(InRaster,InRaster,1.5,"value <= 1.5") will produce a raster where values are as input if they are less than 1.5 and 1.5 if greater than 1.5.
If you don't have spatial analyst you can do Raster to Point, query out the points that exceed the height with Select and then convert back to raster using Point to Raster (use snap raster and cell size of the original raster) this requires an Advanced license. Again this will leave NODATA areas where the value exceeds 1.5, if you want to cap the values then you will need to select and field calculate the values to 1.5 rather than extracting them.
If you don't have Spatial Analyst or Advanced license you will need to download QGIS and use QGIS Raster Calculator because you can't do it in Esri.