You will need to purchase a T-Money card to be able to ride mass transportation in seoul, by buying it at airport's convenience store (CU, GS25, etc)
T-Money priced about 4000 won (doesn't include balance)
Follow the signs to go to the airport railroad and find a machine to fill your T-Money's balance
There will be choices for you on how much balance you want to fill to your card (fyi 10000 won is enough for 2 days trip)
Go to the subway station, follow the signs to the platform of the train to Seoul station (Airport Railroad Line)
Upon your arrival in Seoul Station, just follow the signs to transfer to Line 1, alight at Sinseoldong Station (Line 1) after 6 stops
Get out from exit 11, walk straight until you find Hollys Coffee.
There are three roads. You will see the road on the left side of Hillstate Apartment. Walk down that road and make right turn. Hostelkorea is at second block on your left.
If you plan to use a sim card, you can buy sim card at convenience stores (CU, GS25, etc) for about 30.000 won
Another good choice is you can take airport limousine bus by buying ticket first at arrival halls (1F) price: 10k won
Take airport limousine bus no. 6002 at 5B or 12A Incheon Airport and get off at Dongmyo station
Then cross the crosswalk, and turn right. Walk straight until you come to the convenient store(C-Space).
Go right here for 2 blocks and turn right, you will see Hostel Korea.