I am not familiar with Zip Codes but would imagine it is very similar to the following method for exporting Postcodes (UK) to Excel...

Open New MapPoint Map and Click Data>Data Mapping Wizard (Ctrl + W)
Leave at default of "Shaded Area" then <Next>
Leave at default of "Add Demographics to Map" then <Next>
Select any of the options (eg Area_sq_km) (will be deleted later anyway)

In the lower part of the dialog select the level of info you want
For the UK this breaks down as:-
Postcode Area (eg "G" for Glasgow)
Postcode District (eg "G2" for Glasgow City Centre)
Postcode Sector (eg "G2 4" for Charing Cross, Glasgow City Centre)

Then click on finish and MaPoint will map your selected data to the selected detail level. Once mapping has finished use the drawing tools to draw a line (or circle, rectangle, freeform etc) around the area you want to export (rectangle can be done by left click and dragging on the map).

You can now export the data to Excel (Data>Export to Excel) and it will open as a new workbook. Delete the columns with the Demograpic Data and Country/Region - you are then left with a list of the Postcode Sectors.

There is no way the full Postcode / Zip Codes information could be exported as they would run into hundreds of thousands. With the UK there are 9145 Postcode Sectors alone and each of these has a potential 676 further variations.

Hope this helps

I.T.