There is a hydrofoil service from Milazzo port to Alicudi twice a day run by Liberty Lines, the whole trip takes about 3 hours, Alicudi is the last stop. You can check the departures and prices on the following link:
http://www.libertylines.it/
There is also a possibility to take a Siremar ferry boat which leaves Milazzo port early in the morning most days of the week. The trip takes about 6 hours and the ticket is almost half the cost of the hydrofoil fare. It is a more scenic trip as you can enjoy the views of the other Aeolian islands you stop by – Vulcano, Lipari, Salina and Filicudi. You can check the times and fares at the same Siremar link.
http://www.siremar.it/
Only on Tuesdays, there is a ferry boat service from Milazzo to Alicudi run by NGI company. The same information as for the Siremar ferry applies to NGI trip. You can check the departures and prices on the following link:
http://www.ngi-spa.it/default.asp
During the summer season there is a daily hydrofoil service between the Aeolian islands and Palermo run by the company Liberty Lines.
The trip takes the about 2 hours, Alicudi is the first stop.
You can check the departures and prices on the following link:
http://www.libertylines.it/
There is a also a SNAV company ferry service between Naples and Aeolian islands with ferry leaving Naples usually twice a week. The ferry arrives to Salina where you need to get off and take a Siremar hydrofoil to Alicudi.
In the summer season SNAV also operates a hydrofoil service from Naples to various Aeolian islands (Alicudi excluded).
You can check the departures and prices on the following link:
http://www.snav.it/en/destinations/aeolian-islands.html
You can certainly make a grand entrance to Alicudi by a helicopter :) which can be booked at the following sites: