Upon arrival, your cheapest option is to get to Dubrovnik's Old Town is to take an airport shuttle (@35 KUNA) which picks up passengers outside the arrival hall and drops you off just outside Pile Gate (main entrance to the Old Town). Airport taxis are also available just outside the arrival hall. The taxi line is next to the shuttle bus. If no taxis are available, you can request one from the information desk.
If you arrive by automobile, the city recently completed construction of a brand new parking garage located @15 minute walk from the Old City gates. The Public Garage is probably the best place for you to park. When you drive into Dubrovnik, follow the signs for the Old Town (Hotel Argentina, Hotel Excelsior). There is a very very large sign "PUBLIC PARKING" so you can't miss it.
International ferries from Bari, Italy to Dubrovnik and if rumors are correct, starting in 2010, from Greece to Dubrovnik arrive to the port in Gruz. Long distance buses and local ferries also arrive in Gruz. To get to the Old Town, you can either take a taxi or ride the public bus (15 KUNA). If you buy your bus ticket from the newspaper kiosk, it is cheaper. Hop on either the 1A or 1B 'Pile Gate' bus. Once you get to the Old Town, enter through Pile Gate (main entrance). The drive is @ 10 to 20 minutes (depending on traffic). If you decide to walk, it should take you 45 minutes - 1 hour.
Upon arriving at Pile gate (no. 1 on the map) please call Igor Tošović (00385917946610) and he will meet you or if you like to reach apartment directly just enter Pile gate (the main entrance to Dubrovnik's Old Town), head straight down the main street Plaza (Stradun) and take 5th street from the right (Đorđićeva 3). Once you are in front of the building please contact Igor Tošović on the same number.