I was in Naples for the best trip of my life between 28 Mar - 1 Apr 2019 (i.e. right before the start of tourist season in and around the area). During these 4.5 days, my husband and I extracted as much from the trip as we possibly could. Our biggest challenge seemed to be hiking Mt Vesuvius and Pompeii the same day, without a rush. Here is the itinerary that we managed to pull off-

• Day 1- Land in Naples early morning and see Naples
• Day 2- Mt Vesuvius and Pompeii 
• Day 3- Ischia
• Day 4- Go to Caiazzo to eat the world's best pizza at Pepe in Grani (you will need to reserve a month in advance)
• Day 5- More pizza in Naples before departure at 3 pm

I would not get into the whole trip and rather stick to this post's headline- Yes, it is easily possible to hike Vesuvius and visit Pompeii the same day BUT ONLY IF you start the day early. Upon leaving our Airbnb in Naples at 8:30 am, we got 3 hours to spend in Pompeii after the Vesuvius hike, but we wished we had one hour more. Pompeii is huge so it needs time and another factor is how much of a history buff you are!

Connection details 
1. Naples to Vesuvius 
EAV bus was suspended while we were there and there is insufficient information related to its stops and timings. We were not very trusting of the tours/tourist buses either. So out of renting a car versus train, we picked the latter option.
We planned our train journey before going to bed the previous night, using Google Maps but the next day (and through rest of the trip), we realised that trains in Naples have delays and nothing runs as per the schedule on Google Maps. What can definitely be trusted is the LCD screen at the train station.

Leg 1 of the journey: Naples to Ercolano Scavi
We reached Porta Nolana train station (you can go to Garibaldi as well) and bought tickets for Poggiomarino 8:55 am but it arrived at 9:06 am and departed at 9:13 am. We double-checked with the locals on the train (the majority do not speak any English) if we were indeed headed to Ercolano Scavi. The train did not stop at all the stations and we reached Ercolano Scavi sooner than the 15 minutes ride that it was supposed to be. We had to pay keen attention to the announcement in the train to not miss our station.

Leg 2: Ercolano Scavi to Vesuvio
It was 10:30 am now and our original plan was to walk 40 metres to Panoramica bus stand and take the Vesuvio bus that would drop us at our destination in 30 mins. BUT we did not take that bus because as soon as we walked outside the train station, we saw a 'Vesuvio Express' (tours and travels) bus right in front of us, about to leave. It was a €20 return journey and add to that another €10 for the entrance ticket to Vesuvius National Park (which is the trail that you hike up to reach the crater). This entrance ticket can be bought after reaching Vesuvius as well. The bus returns for the pick up within about 90 minutes. The hike is an easy one, 30 minutes each side, so there is ample time to get the most out of it.
It is very windy at the top, so carry a windbreaker or a rain jacket.

2. Vesuvio to Pompeii
We came back to Ercolano train station with the same bus and boarded the Sorrento train to Pompeii Scavi station at 1:50 pm. As a result, we had 3 hours to explore the ruins of Pompeii from 2 pm to 5 pm (closing time).  3 hours were not enough for us (4 would have been ideal) and then it was time to head back to Naples.
Also, we took the audio tour at Pompeii but I recommend hiring a tour guide for the fun historical details they share. You can hire them at the entrance to the ruins.
Another tip- You could also choose to explore the ruins of Herculaneum (at Ercolano) instead. The site is less touristy and more intact.

3. Getting back to Naples
From Pompeii Scavi, we took a direct train back to Naples called Napoli Porta Nolana - about 40 mins ride, to enjoy more Neapolitan pizza!