Travelling to and from Astypalea requires a bit more forward planning than most other islands. However, things are made somewhat easier by its having an airport with direct flights to/from Athens.
Olympic operates a flight every day except Tuesday direct to/from Athens and Astypalea airport, which is 8km from Astypalea Town/Chora. Your hotel will probably arrange free transport from/to the airport so be sure to check.
Surprisingly, for such a small island, Astypalea has 2 ports. The story is that the locals didn’t like the big ferries calling into Astypalea Town and waking them in the dead of night so the port of Agios Andreas was built, in the middle of nowhere, so that the Astypalians could get a decent night’s sleep. It can present problems for travellers arriving or departing using this new port, particularly given the dearth of taxis on the island.
The large Blue Star ferries to/from Piraeus, via Paros, Naxos, Donussa and Amorgos (Aigiale), terminate and turn around at Agios Andreas, which is 7km from the main town of Astypalea/Chora. During high season, a Blue Star ferry goes non-stop between Piraeus and Agios Andreas continuing on/returning from Megisti via Kalymnos, Kos, Nissiros, Tilos and Rhodes. All these ferries arrive and depart at inconvenient times in the middle of the night. Many of the hotels and studios will arrange to collect and deliver you to/from the port so you should mention this when you book.
A smaller ferry, the Nissos Kalymnos, runs twice a week between Kalymnos and the more convenient port of Astypalea Town/Pera Gialos. Even though the ferry port is much closer to your likely accommodation, you should still try to arrange for your hotel to meet the ferry because of the difficulties in finding your hotel in the maze of streets and alleys and the number of steps you’ll need to climb to get to it.