Yes, it is very easy to go to Atlantis on your own... We had a wonderful day there! There should be taxis waiting. They will want to fill their taxi, so you may need to kind of hook-up with another party to fill the taxi van. We lucked out and another party walked up in just a few minutes.
We found a great rate on a room at the Comfort. Placestostay.com showed promotional rates that we did not see elsewhere. So, check there for your date... (you have to proceed as if you will be booking a room.) Then we called the Comfort Suites 1-800 reservations number directly to book the rate we found ( it was the LGOLD rate ) It saved us a LOT over the cost of a room at Atlantis. It is a simple walk across the street.
The dig is awesome! The pools are awesome! Gorgeous with tons of shady palm trees. The beach that we walked to was not great.... Drop-off a little steep... Surf a little rough... No way DS could get in.... No shade or beach chairs set up for the guests... etc.... You may want to stick to the other areas at Atlantis and skip the beach there.
We had no problem at all going to Atlantis on our own. Had a great day!