On leaving the airport just turn left and head straight for the border which takes about a 15 minute drive. Once in Montenegro drive through Herceg Novi and continue on to Bijela which will take about another 20 minutes. At Bijela you have the choice of driving around the bay taking in the breathless scenery or taking the car ferry across to Lepetane (just follow the signs). This is slightly quicker due to the new tunnel link between Tivat and Kotor.
Podgorica Airport, Montenegro
Podgorica (Montenegro's capital city) airport is approximately 1 hour 30 mins drive from Kotor Old Town and is served by Montenegro Airlines which flies direct from London Gatwick througout the year. Tickets can be booked directly through the Montenegro airlines website or through Opodo. It's also possible to fly with JAT airlines via Belgrade and Paris. Again there are plenty of car hire companies to choose from including Avis, Europcar and www.inmontenegro.com.