Yes, you need the apostrophe, but the whole phrase is rather old-fashioned sounding anyway. Good luck!

like Spencer said, yes the apostrophe is needed in this case because it is not referencing the units of time themselves (day, month, year) but literally the amount of time that these units consist of. To say 'in one day's time' is using a possessive, literally saying the time is owned by or of the day. It is more commonly said as simply 'in one day', here using the unit directly.