Hi, annshandle,
As you'll have Friday to be footloose in Aberdeen, you could possibly visit the Maritime Museum, which is a short walk from the ferry terminal. You could also visit nearby Provost Skene's House, which dates from 1545, and is considered to be the oldest standing house in the city. There are several good places to visit in the city, most of which are outwith the city centre. There are several companies running day tours by minibus from Aberdeen. A coastal tour would be good, or a tour along Royal Deeside.
McGill's Coaches operate the Aberdeen Adventurer, which is a hop on/hop off bus which takes visitors around the historic parts of the city. It might be worth taking the Adventurer to find parts of the city which you'd want to explore in more detail.
You could also take the Stagecoach bus (X7?) to near Dunnottar Castle, which is just south of Stonehaven. Alternately, you could take the train to Stonehaven, and walk or take a taxi out to the castle. If you decide to train and walk, have a look at the impressive WWI memorial on the hilltop before the castle approach. Stonehaven itself is an attractive town, with a beach and harbour. It also has Britain's largest open air heated sea water swimming pool. And is the birthplace of the deep fried Mars bar!
If you're heading south, you could conceivably take the train or bus to Stonehaven, visit Dunnottar, then take ongoing public transportation south from there. Montrose and Arbroath are also good for a half day visit.
There are several good hotels near the ferry terminal in Aberdeen. As you'll be there in May, you should be able to find a room (or rooms) at a reasonable price.
Enjoy Shetland (and Aberdeen)!
Mike (Auchterless)