You should have the 'Print' operation be scripted and driven by a 'Print' button you create on the layout. At the end of the script you can add a Set Field step, where you set the field value to be Get (CurrentTimeStamp).
This will not work if the user just selects File - Print, as that will bypass the script. If you have FM Advanced ytou can modity the menu to remove Print, but you will then have to have a Print button for every time the user would print. (Though not a bad idea, as it saves them having to do the right page orientation, sorts, etc beforehand, or even selecting the correct layout to print.)
Also, if this is a multi-user file then one person's timestamp will over-write the previous person's.
If you have FileMaker Advanced, you can install a modified menu where selecting Print from the Print menu or using its keyboard shortcut performs your print script--making this all transparent to the user--they think that they are printing like they always do, but your timestamp field is magically updating each time that they do.