Here is the code to do as you ask.  If you need help adding new toolbar
buttons and assiging macros, please see the standard Word help menu (not the
VBA help) and look under toolbars, add buttons....

Switch to the non-default printer to which you wish to do the printing.  
Then, the VBA mode use Jean-Yves' code regarding determining the proper name:
? application.ActivePrinter (It's Ctrl+G or View, Immediate Window)

Copy the name (don't include the ending colon), and then switch back to your
default printer.

Dim sCurPrinter as string

sCurPrinter = ActivePrinter  'This is your default if you haven't switched

   ActivePrinter = "plug in the text you copied from the immediate window
between the quotes"

'If you want to control the paper bins, you can do something like this
'Note: bin names are printer-dependent

   With ActiveDocument.PageSetup
       .FirstPageTray = wdPrinterLowerBin
       .OtherPagesTray = wdPrinterLowerBin
   End With

'If you want the user to be able to print only parts of the document,
'or multiple copies, etc., use the following code.  User will click send doc
to print

   Dialogs(wdDialogFilePrint).Show

'Or, to just print the document without user-intervention, use this code:

Application.PrintOut FileName:=""

ActivePrinter = sCurPrinter 'Switches printer back to your default

'You might want to find out the name of the default printer just to be sure
and use
'that instead of sCurPrinter (again, use Jean-Yves' immediate window code to
'do so)