Kirkwall would be the best centre for both public transport and any tours.
Stagecoach run the bus services on Orkney . Have a look at their website which gives a guide to what places you can visit by bus. You will need to spend time checking out bus times and whether they work for you.
This is the general information about using buses from the Orkney Islands Council website.
WOW Scotland do day tours of Orkney from Kirkwall, but I’m not sure if these are just geared up to Cruise ships. Their website says they aren’t doing tours at the moment. It may be worth contacting them and asking. They would also be able to give an idea how many people there are on a tour and how long you spend at each site.
See Orkney are another possibility and their website says they offer small group tours. They may be a better bet?
You can also find out information about private tours here, which would be tailor made to your specific requirements (but will probably be much more expensive) . There is information about a tour of the upper levels of St Magnus Cathedral too
It is difficult to say how many days until you have sorted out your priorities and how you are going to achieve them... Off the top of my head I would suggest a minimum of three days and ideally longer. Having made ther effort to get to Orkney, you don’t want to short change the islands!
Use the map on the Undiscovered Scotland website to check up what things there are to do and see once you arrive. Click on the names for text pages with lots of information and pictures. Then use google to get latest information about opening times etc.
The Orkneyjar website is another wonderful resource covering the heritage of the islands and covers lots of topics not found in the usual guide books.
If visiting Culloden, don’t miss Clava Cairns which are close by
Where are you catching the ferry from? Are you using North Link from Aberdeen leaving Aberdeen at 5pm and arriving at Kirkwall at 11pm? Accommodation providers are well geared up to these late arrival times.