There are two ways to get to Ærøskøbing from Sønderborg and neither are fast. The first is just the continuation of the journey already mentioned via the ferry from Svendborg.
The second is via the ferry from Fynshavn on the Als island. However this ferry goes to Søby on Ærø, which is the other side of the island from Ærøskøbing and there does not appear to be a bus that goes all the way from Søby to Ærøskøbing.
You will need a bus from the busstation in Sønderborg to Fynshav, a ferry from Fynshav to Søby, a bus from Søby to Tranderup on Ærø and probably walk the rest of the way (5 km). Or just see if you can get a taxi from the ferry in Søby.
Use www.rejseplanen.dk to plan all public transport in Denmark. However note that since there is no connection between Søby and Ærøskøbing, it won't show you the route via Fynshav.
The ferry scedules to Ærø can be found on www.aeroe-ferry.dk