I have taken a taxi there before, of course it can be done. However, Rio can be a dangerous city. Dress down and leave the Rolex at home. Don't carry an expensive camera AND a big bag that looks like it is full of expensive things.
I am the editor, but I also speculate, ask questions and play devil's advocate. I reserve the right to change my mind.
Yes, you can do Ipanema on your own. Take into consideration what Paul said and only take the essentials and go during mid-day. Stay to the beach area and you'll be fine but just use common sense like in any large area.
A cab will run you about $25 from the cruise ship port that Azamara uses, or at least did on our December cruise.
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"There is a great difference between being well traveled and just having been to many places." ~Me