Question about Lima Int'l Airport

Hi

I will fly to Lima and arrive at mid night and I know it is dangerous to go out.

I want to go to Cusco next day. My question is

If I go by bus, can I sit down somewhere inside the airport and wait until next morning then take a taxi for the bus terminal?

Or if I go to Cusco by plane, is it the same airport ?

Please help me

Thank you

If you go to Cuzco by plane, it's the same airport (int'l airport)