The fastest, easiest way to travel to and from Astypalaia is by plane.
Astypalaia has got an airport and there are direct flights from Athens daily, except Sunday.
However, if travelling by ship is the option, the following choice of routes, if convenient to your general travel plans, can minimize any potential port delays and avoid huge crowds on board - so try to sail:
Athens/Piraeus to Astypalaia with Diagoras, which sails direclty to Astypalaia, its her first stop and then other ports follow
and on the opposite leg, from Astypalaia to Piraeus get the Blue Star Paros which, again sails first from Astypalaia and reaches several other ports before arriving at Piraeus.
This way there will be no delay on the trip to Astypalaia - no other ports/stops in between
and on the way back finding a place to sit/rest for the journey is guaranteed....
Full itineraries can be found following the link below
There are two busses that run along the island and their destination and times are on a board at the bus stations. They are very inexpensive but beware sometimes the bus driver thinks he is related to Schumacher and drives like a maniac. The busses run along the paved roads between Hora, Pera Yialos, Livadhia and Analipsi.
There is another ferrie that calls on the island twice or three times a week, but it only takes you to Kalymnos from where you then have to catch another ferrie to where ever you wanted to gho and the people on Astypalaia could not tell me where to buy the onward ferry ticket or how much it costed.
Flying to Astypalaia is prob your best bet though prob not the cheapest. I paid 50 euro for my flight, but it was well worth it due to irregular ferry visits and/or ferries arriving middle of the night.