Have you considered going to Ostia Antica instead of Pompeii? It's an easy 1/2 day excursion from Rome by local train. Save Pompeii and Naples and so much more in that region for your next trip.
On April 22, sunset in Naples is at 7.20 pm. According to the official website, Pompeii is open daily until 7.30 in April. I would have said you will have time to visit the site and also the museum in Naples, but then I read one of the other responses that mentioned kids - so I looked for other threads of yours and found that you will have two children, aged 6 and 11, with you. That makes a huge difference for a very long, strenuous day
A day trip means getting up and out early.
0.5 hours from hotel to train station
1.5 hours to Naples (I added a few minutes to find the train and board in Rome)
0.25 hours to get to the Circumvesuviana train and buy tickets
0.45 to Pompeii (could be longer due to wait time - only 2/hour)
So that's about 3 hours until you get to Pompeii. The site is huge and even if you don't visit the outlying areas (I don't think the amphitheatre is worth the time and it's a really long walk to the Villa dei Misteri), you'll need several hours, including a lunch break. There are few places to sit and rest and walking on ancient cobbled streets or very narrow sidewalks is difficult in some places.
You'll want to allow enough time for the museum in Naples. The Circumvesuviana schedule isn't reliable (trains are often late). I don't remember how long it took to get to the museum from the train station. As I recall, it's a longish walk from the CV to the metro, but a short walk from the metro to the museum. There's a lot to see from Pompeii and the other excavated cities and other fine collections of ancient art and artifacts. With perfect connections, it's still roughly an hour of travel time. From the museum it's another 2 hours back to your hotel.
So at least 6 hours of travel, and an hour for lunch. A 12-hour day leaves you only 5 hours of sightseeing time and all of it on your feet. It's not easy for adults. How are your kids going to deal with it. Make sure they use the toilets when you pass them - they are few and far between. If you want to add a stop for snacks, and maybe dinner in Naples, you will either have less sightseeing time or an even longer day. If I were 6, I'd be cranky long before I got to the museum and if I were 11, I'd be either unbearably whiny or moodily plotting revenge. Hopefully your kids are better travelers than I was as a kid.
Lastly, most people by their train tickets in advance to save money. The full adult fare is €23 more than the super economy fare. so for 2 adults, the difference is €90. That reduced fare is often less than the discounted child fare also. (caveat - maybe there are cheaper tickets for round-trip or family fares). That means deciding in advance not only what day you go but also what time you leave and how long you stay.