I highly recommend these apartments. We stayed in 'Lapad' in the Adria Apartments and it was lovely. The rooms are a good size and the air conditioning is highly effective. The unit is situated in the bedroom but it is quiet and on the lowest setting will keep you cool (but not cold) all night long. If you leave the door open, it will easily cool the whole apartment, which is needed in the heat of high summer.
The apartment is easy to get to from the Airport and from the Old Town. From the Airport take the Atlas Bus to Pile Gate in Dubrovnik (30 mins) (c70-100Kunas pp / £9-£12pp), & then No. 6 Bus (Baba Kuk) to Lapad Post Office, which takes 10 minutes (10Kunas pp/£1.20). The apartment is then a 5 minute walk.
Taxi direct to the apartment is 250 Kunas. The flight is likely to land early (10am) and the apartment is not ready until 2pm so get the bus to Pile Gate and have lunch in the wonderful Dubravka Restaurant with views of the sea, city walls and fort. Then get the bus to the apartment.