Hello Ranah, and welcome aboard! I don't blame you for wanting to see as much of Atlantis as you can. It's truly an amazing place!
I haven't been on that specific Port Adventure, so I checked with Disney Cruise Line. Unfortunately, the Discover Atlantis adventure does not grant access to the pools, water slides or beach at Atlantis. Only what is stated on the website.
The
Atlantis Beach Day Port Adventure, however, does grant you access to the spectacular beach, though the cost is more than the other adventure you mentioned. Because Atlantis is a resort unto itself, only their guests have full access to all that it has to offer, unless you book a specifically arranged adventure.
You might look into walking about 15 minutes from the port over to Junkanoo Beach. You can rent chairs and umbrellas there, and this could save you some money if you decide not to go to Atlantis. There you will find local food and music. It's not quite as quiet and relaxing as the beach at Atlantis, but it's close and can be a fun spot to spend a few hours.
I hope this helps in your planning, Ranah.
Have a wonderful cruise!