To do this in ArcMap an Advanced level of license is required.
First break down the polygons into lines using "feature to line"
If the lines aren't broken into segments properly use "planarize lines" in the editing toolbar, this will give you short line segments mostly two point lines which should be acceptable depending on how picky you need to be, if you need to you can manually split some using the editor "Split tool".
Save and stop editing. Either add fields using attribute table or "add field" tool, all fields are type double. Add X_From, Y_From, X_To, Y_To and Orient fields, open the attribute table in ArcMap and populate with "Calculate Geometry"; populate the X_From as the X Coordinate of Start, Y_From as Y Coordinate of Start, X_To as X Coordinate of end and finally Y_To as Y Coordinate of end.
While the attribute table is open calculate the field with a python code block using the formula:
(math.atan(( !Y_To! - !Y_From! )/(!X_To!- !X_From!)) * 180)/math.pi
This will give degrees, some will be negative, depending on how you want your angles stated you might need to do some Pythagorean maths here.
Now that the angles are populated on the lines use "Near" to determine which is the closest line to each point. This tool will add a field called NEAR_FID (or something like that) to the point feature class with the FID/OID of the closest feature in the lines. Join the table of points onto lines using NEAR_FID to FID/OID and then calculate the orientation onto the points.
There are quite a few steps involved but it can be done manually; it is this sort of convoluted process that encourages aspiring GIS professionals to learn Python! Please consider doing some of the on-line tutorials in Python for ArcGis, it will make your tasks much easier and less reliant on remembering.