Southampton > Portsmouth Transportation Options

You have two options, bus or train. National Express Bus leaves Southampton roughly every two hours, takes about 1 hour, and costs starting at 5 pounds. https://www.nationalexpress.com/en
Train. Up to three trains an hour. Takes around the same amount of time depending on the service, and costs start at 12.6 pounds.
https://www.nationalrail.co.uk/

Once you get to Portsmouth harbor, it is very easy to do sightseeing on your own. The historic shipyard is just a few minutes from the station. I'm going to be doing this same thing this spring where I stay in Southampton prior to a cruise and take the train down to Portsmouth for the day. Both train and bus are good options, and you may want to see which station is closest to your hotel.

There is a third option, a private hire car provided by AJS Travel https://www.facebook.com/profile.php?id=100063321524226

I know the drivers personally and have been using them for all my airport transfers. I've recommended them on this forum on a number of occasions and the feedback has been positive. The drivers are all Portsmouth residents and will happily provide advice and guidance.

There is also a local citybus service- the X4- runs hourly. Takes 2 hours, usually run with double decker buses. £2 each way.

If you leave after 12 noon the train is only £10.50 return with what is called an 'Evening Out Return'. This is a walk up ticket, yes I know the evening does not start at 12 noon, but it does on railways in southern England.
Leaving Sou Between 9 am and 12 noon it is an Off Peak Return at £13.80
Before 9 am it is an Anytime Day Return at £13.90