Can I visit Machu Picchu on the same day I fly to Cusco?

Yeah. The first flight from Lima to Cusco is at 5 am. Arrive in Cusco at 6.20 am. From the airport to the Poroy train station takes about 45 minutes by car. And there are direct services to Aguas Calientes (Machu Picchu Pueblo) such as the luxurious Belmond Hiram Bingham that departs at 9 AM and arrives at its destination at 12:30.

You can enter the Inca city at 1 or 2 PM and take advantage of the less tourists in the afternoon visiting the citadel of Machu Picchu.