By bus: head to Spiaggia delle Monache to find free-access beach or rent umbrella and sunbeds from at Bagno Sirena , or keep walking to the neighbours Bagno Elena and Bagno Donn'Anna
At Spiaggia delle Monache, consider to book a canoe trip along the shore. Unforgettable. https://www.kayaknapoli.com/
If you are have a car, baiadellerocceverdi.it is another option. Prices on the the website
La Gaiola is an unforgettable spot but getting there is a bit more adventurous. It's a 40 minutes walk, but you can drive up to very close and walk about 10 minutes!
During summer entrance is free but a reservation is necessary.
Book your access through the website https://booking.areamarinaprotettagaiola.it/booking/
There is no shortage of options in Posillipo, most of them have been in business for at least 50 years and are at walking distance from BayView Posillipo.
Walking eastbound (Vesuvio on your left-hand side):
"Al Poeta" is in the nearby Piazza Salvatore di Giacomo, literally 5 minutes walking
"Pruneto 1944" is another option, about 20 minutes of breathtaking views
Walking westbound (Vesuvio on your right-hand side):
Walk 15 minutes to Piazza San Luigi and stop at "Elettroforno Giulia" for a take-away slice: it may look basic, but be aware that crowds of youngsters gather here from all around (driving for an hour or more) to eat street food while looking at the view.
Several other options exist in the area, and all worth a visit
Are you driving a car?
Can't miss to visit "A' Fenestella" to eat pizza literally "at sea"
Watch out: traffic restrictions may apply!
Again, there is no risk of disappointment!
For the best "gelato" (ice-cream), walk again to Piazza San Luigi and stop at the famous gelateria "Bilancione", considered Napoli's best by many locals.
"Pasticceria Napoletana" is also as a close as a good spot to taste pastries
The view from Parco Virgiliano is perhaps the most stunning you can get all around Napoli: see the entire bay, both to the South (Vesuvius, Sorrento, Capri ...) and North (Pozzuoli, Procida, Ischia ...)
The perfect spot for a sunset drink