With the exception of some abilities on characters entering battle, ability order is print order. So, if you activate it, each player does the hand-check and discards if needed. Then the holder does a Roman-check and draws if needed.
My question regarding this card and "when it is activated": If I leave Denarius up from a previous turn, does it 'activate' immediately upon my Prep phase starting, or can I lay down and say "now it is activating". This is an interesting case, as existing rulings state that you cannot deactivate a previously activated artifact, perform actions, and then reactivate that same artifact. However, those are all rulings on ongoing abilities, not instants like on Denarius. For artifacts with "instant" abilities upon activation, does it activate at any time the holder chooses during Prep Phase, or does a previously active artifact go off at the beginning of Prep Phase if not deactivated?
Regardless, this points to more need for an Artifact Phase... ._.