By car. Yes you can easily do this in a day.
It's easily possible by train, as long as you're OK with a fairly early start....
On weekday mornings (didn't check Sat or Sun) there are a couple direct from Roma Termini to Caserta - at 08:05 (68 minutes, €32 in 2nd... less if lucky) and then 08:14 (2hrs 22 mins, €23.50).... although the next, at 12:35, is a slow (but cheap, €12) Regional service, which takes almost 3 hours - otherwise it's down Naples and change there to go back up again!
But at least the station could hardly be any nearer the Palace?
(I imagine that those works, photographed in 2010, must be complete by now!)
Haven't tried, but am fairly sure they've no left luggage facility there - will check re one at the Palace, if you want?
Then a fairly good service from Caserta to Naples, taking 40 to 50 minutes and costing only a few Euro - with tickets for those bought locally...