There are a lot of ways to reach the paradise. The easiest one is to rent a helicopter (we have a heli run way) for a lot of money, which you could better spend in Ouzo, make your choice.
Well, back to basics.
Arki is connected with the metro line 3 from Athens. Don’t tell me now you believe that, do you?
Well, back to basics two...
The nearest island that has a direct itinerary from and to Piraeus is Patmos. In the summer months there are daily trips. The timetable varies according to the company. The journey, Piraeus-Patmos, is roughly 9 hours taking into consideration that Patmos is connected to Arki with tourist boats. The boats depart daily at 10 am. It would be better to choose a night route so that you arrive at Patmos in the morning and then you can catch directly the boat that will take you across to Arki. Alternatively, if you find a morning itinerary you can stay over the night in Patmos and get on the boat to Arki the next morning!
Flight to Samos or Leros and from there you'll have to take the `Nisos Kalimnos', a local boat that begins its journey every Monday-Friday-Sunday from Kalymnos in order to reach Leros and then Lipsi, Patmos, Arki, Agathonisi and finally Samos. From Samos it leaves again at 2.30 at midday in order to reach again Kalymnos. Thus if you find a convenient flight to Leros or Samos, keep in mind that you'll leave from Leros at 7:00 am, or from Samos at 2.30 pm.
Because we live in Greece and the itineraries may change without any warning, it would be better to contact a travel agent first before booking your tickets.
Here are some links that you will probably find very usefull: