I can't help with Uluru, but I visited Cairns and Port Douglas as a solo traveler in 2019. That's becoming a while ago now, which I is why I mention the timeframe. Things can change, but current options can easily be researched. I'm just going to to give you a brief run-down of how I used my time in Cairns and P.D. since we seem to have similar interests and timings.
1) I wasn't eager to rent a car and managed fine with public transport options. A car obviously gives a bit more flexibility, but I found the options for getting around without one didn't restrict anything I wanted to do. I know many do rent cars - just not my preference, so relating my experience for any readers who are hesitant.
I found that there were a number of convenient bus shuttles, for instance to Kuranda Skyrail and Hartley's both from Cairns and P.D. I booked a shuttle between PD and Cairns - inexpensive and a shuttle from the Cairns airport to P.D. where my first lodgings were. Day tours to Draintree were also abundant, similar, and all well reviewed. The one I chose was excellent. I checked with several companies - back then, they promised to run the tours even if I was the only person registered. That isn't always the case for solo travelers.
2) In Cairns, for the "chill day", I enjoyed the Botanic Garden and Aquarium that focuses on local aquatic life - fresh and salt water. I Ubered to the Botanic Gardens from my hotel on the coast. Cairns night market - especially if new to night markets such as found in SE Asia - might be worth a stop.
3) I did one snorkel trip from a P.D. company and another from Cairns. I'm a certified scuba diver, but didn't scuba the GBR, just snorkeled. My observations of the scuba option from snorkel boats was: if you are a certified, experienced diver, find a dedicated scuba trip, not a tourist snorkel boat. The trip from Cairns was better than out of P.D. - the P.D. reef site was quite bleached. Be careful of the boat you select - some of them are huge with large tour groups. Lots of inexperienced, even non-swimmers on these snorkel boats and even first time scuba participants.
4) In P.D., this is an attractive walking/hiking trail leading to (or from) 4 mile beach. https://douglas.qld.gov.au/parks-reserves-and-foreshore-hire/flagstaff-hill-walking-trail/ Watch for crocodiles on 4 mile beach (seriously).
5) From Cairns, some people enjoy tours to the Atherton Tablelands - foodie sort of stops. This isn't my interest, but I did find a driver who took me to some of the sites I was interested in - nature lands and some historical sites. Just mentioning in case you haven't looked into that area. https://www.athertontablelands.com.au/destination/atherton-tablelands/atherton/
The distances in that area - P.D., Cairns, Draintree - are not bad. Roads are good, barring bad weather. I wasn't there during your timeframe, so don't know what you might encounter. I spent the same amount of time you are contemplating with 2 bases - Cairns and P.D. I felt the timeframe and two locations gave me a great opportunity to see all I wanted at a relaxed pace. I hope you have a great time - I certainly did!
