You should be able to move out from the current apartment at the midnight (just before 0:00 sharp) on the last day, and move in the new apartment anytime after still 0:00, unless your new lease starts other than the 1st day of the month.
That's why you see many people moving in & out during the last day of each month - quite commonly. So, talk to your new landlord to tell them you have to move in on the first day; otherwise, you will have to stay in hotel and they need to cover the costs.