You'll have to do this manually each time you play a track, as there are no triggering mechanisms for running some expression to update a field (such as end of track playback).
If you're interested in doing this manually, an expression can be created that you can paste into a field, to take the current field's values and prefix/append today's date. You could do this at one shot for all tracks at the end of playback of a playlist, for example.