Does the tappan zee bridge charge a toll both ways?

No.
There is a toll eastbound/southbound only:

$5.00 (cash)http://www.answers.com/tappan%20zee%20bridge#cite_note-0
$4.75 (http://www.answers.com/topic/e-zpass)http://www.answers.com/tappan%20zee%20bridge#cite_note-1