It sounds like there is a problem with the spatial reference of one of your layers. Actually, it is more likely a problem of the spatial reference you assigned to the points, being different than the coordinate system the points were actually exported in.
Let me explain. You exported a polygon to a shape file. When you did that, you should have assigned it a spatial reference at the time this occurred. It seems like it is a projected coordinate system of some sort. You should be able to find out by looking at the shapefile in Windows Explorer or in your file browser in Linux and looking at the ".prj" file associated with the shapefile.
When you converted the vertices to points, you were probably asked whether you wanted the points to be in the spatial reference of the original polygon shapefile, or of the data frame that you were working in. Whichever one you chose, is the one that you would assign when you added the features as an x,y layer in ArcMap.
For example. The polygon shapefile has a spatial reference of UTM Zone 36N. The coordinate system of your Data Frame is WGS84 Lat Lon. When you converted the vertices to points, you chose the polygon shapefile coordinate system of UTM Zone 36N. When you added the points as an X,Y Layer, you selected the WGS84 Geographic Coordinate System. At this time, ArcGIS is going to take the coordinates that are in UTM and treat them as Latitude and Longitude, which is why the points are landing far outside Egypt.
Say you had a point with a coordinate of E 685000 N 970000
The easting would be divided by 360 for the Longitude, which equals 1902 with a remainder of 280. The 280 would be the final Longitude Value.
The northing would be divided by 90 for the Latitude, which equals 10777 with a remainder of 70. The 70 would be the final Latitude Value.
Please note that this is an example, and the coordinates that you ended up with probably indicate a different coordinate system was used in the polygon vertice coordinate conversion. Once you determine that coordinate system and apply it to your X,Y features, they should line up.
Based on our discussion, it looks like there was some problem in the conversion of coordinates from the polygon vertices to your database table. I think an easier way of fixing this would be to use the "Feature Vertices to Points" tool under Data Management toolbox to do the conversion. This will take the Input polygon featureclass and create a new point featureclass, in whatever format you determine, whether shapefile, or geodatabase, that contains a point representing each polygon vertice, along with a field referencing the feature ID of the polygon that the point came from. Then, you simply add the new featureclass to ArcMap and it should line up. There is no need to use the Add X,Y Layer. This does require an ArcINFO license, but it looks like you might have that.
Alternatively, using the existing table, and assuming that your points are shifted far to the north, do a calculate on your Point_Y field and divide the values by 10. This will take them from x.xxxxe+006 to x.xxxxe+005, which is more in range. Then try the Add X,Y feature.