I'm wondering what forms of public transport exist to get say, between Cairns and Port Douglas / Oak Beach around 4am in the morning? Or will it be necessary to rent a vehicle?
To look up public transport options in Queensland, sadly there isn't currently one single place to look. For south east Queensland, the Translink website has a journey planner that covers buses, trains and ferries for the area. Unfortunately, you're after information on a different bit of the state, so Translink is no help...
Qconnect provides urban bus information and planners in most of the state, which will help you with planning journeys around Cairns itself. It doesn't cover long distance services, which is what you'd want.
The main long distance bus operators on the east coast are Premier Bus (only runs from Cairns south) and Greyhound Australia, but neither seem to currently serve Port Douglas. There are a few local services, such as Country Road Coaches, Port Douglas Bus and Krystal Transport, but nothing that runs that late/early.
The upshot is that I think in order to get there then, you might have to get a taxi...
Or, why not just get the last bus there the evening before, and kip on the beach?
According to the WikiTravel article on Port Douglas there are at least two companies offering shared and group transport between Cairns and Port Douglas:
Niether specifically mentions Oak Beach, but this is a tourism-driven part of the country and airport shuttles are generally extremely flexible with where they will pick up and drop off. But since you are a group it might work out cheaper than renting a car to use one of the group services these companies also offer.
You might even find they have planned especially for the eclipse. I highly recommend contacting both companies. See if there are special eclipse even services, find out if the size of your group is a good fit for a private group transfer, and compare the prices to car rental.
Exemplar meets every flight and offers seats in a coach for $34 per person and charter vehicles for 7 people for $210.
Sun Palm has a regular schedule beginning at 3 a.m. and offers seats in a coach for $35 per person. They require you to contact them about event and group transport so I can't include a price here for you.
If you go for a group charter you might even find the price slightly lower for the shorter distance to Oak Beach. Haggle!