Sunset is the best time to visit Burj Khalifa as the sights are better and it's best to reserve the tickets beforehand so that it's faster to get up to the observation deck.
The Burj Khalifa doesn't close and stays open for 24 hours per day 7 days per week.
The cost to go to the top of Burj Khalifa is $41.00 USD or 149 AED for Adults and Children 12 and above and children under 12 is $31.00 USD or 114 AED.
You can go inside the Burj Khalifa building.
Only the floor to the observation deck is open to the public as well as the other floors on inside the Burj Khalifa.
The only floor that is not open to the public inside the inside the Burj Khalifa which is the very top floor which is the 163rd floor which is not open to the public.
So you cannot access the very tip top of the inside the Burj Khalifa unless you work there or have a reason as a worker to go to the very top.
But all other floors and the observation deck of the building are open to the public.