Is there any left luggage counter near Pompei site, arrive by domestic flight on May 20

Pompeii (in my opinion) will take 3 full hours to see and be able to enjoy/appreciate. Unlike Museums, Pompeii is an actual city (a rather large one) and the interesting sights are scattered apart from one another across the expanse of the city.

I believe they close Pompeii around 6 or 7pm. With your 4pm arrival time (I'm assuming you're coming on an international flight?) adding customs processing, and travel time by bus, train or Taxi...it would be nearly impossible to see or enjoy anything at Pompeii in what little time before sunset remains. The train station outside the entrance to the ruins is really just a train "stop" and to my knowledge does not have a place to check your luggage.

I'd recommend going from the airport to your hotel in sorrento for a nice dinner and gelato and then scheduling a visit to Pompeii the next morning. You'll be glad you did. :)

I agree, you will probably have only an hour if you try to visit Pompeii that afternoon (depending on season and sunset time). Better to visit the next morning, unencumbered by luggage.

You don't have enough time. Once you get off the plane, collect your bags, go through customs, and then drive out to Pompeii, it will be closing. Stay the night nearby, see Pompeii the next day, and then go to Sorrento. Pompeii is an extensive site that you can't see in less than 1/2 day.

I agree with the others that your plan to visit Pompeii en route very late in the day, after a long flight, with baggage, is a very bad idea. Go directly to Sorrento, and then during one of your days in Sorrento visit both Herculaneum and Pompeii by train, with plenty of time to explore and without baggage. I recommend selecting one of the guides near the entry at Pompeii, for a 2-hour walking tour. They know it all, including what areas, streets or rooms may be closed that day for some reason. I tried to visit with just a book and audio-tour and was often lost and missed some key sites. To get from Naples Airport to Sorrento, I understand there is also a direct Curreri Viaggi bus, or, depending how how many people and bags in your group, and how well the bus schedule meshes with your arrival, a nice splurge would be a taxi or private car to take you directly to your hotel in or near Sorrento, probably about 100 euros door-to-door, but you will then have the evening to dine and stroll through beautiful Sorrento. If you benefit from this guidance, then please toast me with your first limoncellos in Sorrento.

Another vote for skipping Pompeii on the date of arrival. You won't get there on time. If you travel from November to March the last entry is at 3:30pm. If you travel from Apr through Oct the last entry is at 6pm.
The Curreri bus ride is about 30 minutes long. The first departure, which potentially could get you there before 5pm (however that's peak time in Naples), departs from the airport at 4:30pm. It would be very hard for you to get that one as you only have 25 min. to deplane, retrieve luggage at the carousel, get oriented and go outside to catch it (passport control is something you will have done at your first port of entry in the EU).
If you miss that 4;30pm bus, the next one departs at 7:30 pm, which obviously will get you there beyond closure time. So your only option is probably to go straight to Sorrento (where you will get on that same bus at about 8:45pm). Then the next day, unencumbered by luggage, you can take the Circumvesuviana train from Sorrento to Pompeii with plenty of time to visit the site in leisure (it really takes at least half a day to visit it all).

Both the Circumvesuviana train station and the main entrance to the Pompeii ruins have bag storage. But I would not attempt this plan. In summer, last entry to Pompeii is about 18:00 and last exit at 19:30; you would feel very rushed.

Thank you very much for your great reply!!! I will go straight to the hotel and visit Pompei next day!