The Vatican Museums has announced that once again this year, for the tenth season, the Museums would be extending their hours on Friday and Saturday evenings until October 28th. For the remaining part of April, the Museums will remain open until 22:30 with last entry at 20:30; from May 6th onwards on Fridays the Museums will remain open until 22:30, but will close at 20:00 on Saturdays with last entry at 18:00.
Tickets to “I Notturni” or “By Night” can be purchased online here, where you will also find all information relative to dates and prices.
If you should want to start your visit with an aperitivo and extensive snack buffet in the magnificent Cortile della Pigna or Pinecone Courtyard (named for its colossal Ancient Roman statue of a pinecone located at the center of a double-ramp staircase designed by Michelangelo), scroll down to Happy Hour and book there. The full-price ticket is 40 euros per person for up to five participants.

An additional and free-of-charge pleasure this year will be concerts on every other Friday evening: April 28, May 12 and 26, June 9 and 23, July 7 and 21, August 4 and 18, September 8 and 22, and October 6 and 20. Beginning at 20:00, they will be held in the Gregoriano Profano Museo of ancient Greek and Roman sculpture and each one will last one hour. The performers will be Italian military bands and student orchestras, soloists, and singers from conservatories in Udine, Campobasso, Ravena, Novara, Cremona, Cosenza, Siena, Gallarate, Torino, and Avellino.
Note that a visit to St. Peter’s Basilica is not included with this ticket. Another word of advice: Book soon because some dates are already sold out.
I’d choose to go on a Friday evening because you have more time. Upon arrival I’d start by going straight to the Sistine Chapel and Raphael Rooms, which are the farthest away from the Museums’ entrance/exit and then work your way back via the not-to-be-missed Map Gallery to the Cortile della Pigna for an aperitivo at 19:00. The Cortile is not far from the Gregoriano Profano Museo for the 20:00 concert, not performed on Saturdays.
Afterwards you will have another good hour to explore the nearby masterpieces of ancient Roman sculptures. To name a few, the Apollo Belvedere, much admired by the famous 19th-century German archeologist Johann Joachim Winckelmann, the Belvedere Torso, which inspired many of Michelangelo’s sculptures, and the Trojan priest Laocoön and His Two Sons being attacked by giant sea serpents.