travel from Heathrow to Portsmouth

You could take the National Express coach from the airport. Google for schedule and prices.

https://beta.nationalexpress.com/en
National Express says it is about a three and a half hour journey.
To go by train is more complicated (several changes) and longer by about a half hour according to National Rail.
http://www.nationalrail.co.uk/
Both the fort guarding the harbour and the still-functioning Royal Navy yards are top tourist attractions. Some of the museums have admission charges. The price to visit the beautifully restored HMS Victory, flagship for Lord Nelson, is stiff but worth it, particularly with a convenient and informative audio guide. The yards are also the damp home for the Marie Rose, risen from the deep.
https://en.wikipedia.org/wiki/HMNB_Portsmouth
http://www.historicdockyard.co.uk/

Regarding the Dockyards, individual attraction tickets are steep, but for the cost of less than two of the attractions, you can get a year pass that covers all attractions. Certainly you won't need the year admission, but it is nice since if you really enjoy Naval history, it will take more than a day to see everything, and the pass helps you spread it out

The cheapest and easiest method will be by National Express coach. You could arrange for an airport taxi service which should cost around £80 for a Portsmouth based. There is no direct train service, you would either have to make your way by train into London and take another from either Waterloo or Victoria or take a bus to Woking and take the train from there.

There are a couple of Premier Inns in Portsmouth, a Marriott that is on the outskirts and not particularly convenient, a Holiday Inn which is in a good location and several independent hotels of mediocre quality. There are many B&B's, particularly in Southsea. I have no recommendations as I've never had the need to stay in any of them.

You will probably be sleepy, jet lagged, and in totally unfamiliar territory. That being the case, the easiest (but not the cheapest) will be a private car service. There is nothing like having a private driver meet you inside the terminal and whisk you to your final destination. No struggling with luggage, no having a lost feeling, and no having to work out the logistics yourself. The convenience, in my opinion, far outweighs the cost -- and it's probably not that high in relation to the cost of the cruise. And often it is almost as inexpensive and public transit for two. It's one of the "indulgences" we allow ourselves. We've had good experiences using car services based in our destination towns as they always seem to know the best way "home". Do some google searching for "Private Car Services in Portsmouth U.K." and see what the possibilities are.

Can't recommend a hotel (I use the 'My Sister's Couch or Living Room Floor' free hotel) but I will say that the Royal Garrison Church was quite interesting (especially if the docent(s) are around) and someone mentioned the Historic Dockyards - give a check online to see if they have any specials on tickets - we lucked into a buy one get one half price on a weekend - we'd kept putting it off because of the price, but that convinced us to finally go and we quite enjoyed it.

We used www.aquacars.co.uk, based in Portsmouth, and the driver was waiting for us at Heathrow after we cleared customs and immigration. The price was less than two train tickets plus we were delivered to the door of our hotel so no dragging luggage on and off a bus or trains. Check it out. We stayed at the Sailmakers Loft, right on the water for two nights. There are several hotels nearby but we wanted a water view from our room and I think we had a rooftop view as well. Check the website and book it quickly since there are only several rooms if I recall. Good Luck.

Thanks everyone for taking the time to share your expertise and advice. Just the type of information we needed. Thanks again