If you want to visit Montenegro by bus/coach, there is many destinations that are offering direct lines to Montenegro. If you are coming from Serbia (or passing through) there are many direct bus lines for Montenegro from Belgrade, and all other major cities.
From Bosnia and Herzegovina there are direct lines for Montenegro from Sarajevo and Mostar. During the summer season there is a direct line from Podgorica (capital of MNE) passing through most of coastal towns of Montenegro and Croatia, and going to Ljubljana, capital of Slovenia.

Transfer from Montenegro to Dubrovnik and vice versa

There is several bus lines between Dubrovnik and Montenegro. “Autosaobracaj” is a company that has daily lines from Dubrovnik to Herceg Novi (Montenegrin town, close to the border of Croatia). From Herceg Novi there are many lines during the day for all other destinations inside of Montenegro.

Autosaobracaj has regular bus line from Herceg Novi to Dubrovnik (daily).

Time table :
Herceg Novi - Dubrovnik : every day in 9.30 AM (main bus station in H.Novi),
Dubrovnik - Herceg Novi : every day in 15.30 PM (main bus station in Dubrovnik).

Also, Autosaobracaj have line from Podgorica to Pula (every friday).

Time table :
Podgorica - Pula : every Friday in 16.30 PM (main bus station in Podgorica),
Pula - Podgorica : every Friday in 16.30 PM (main bus station in Pula),
Pula - Podgorica : every Sunday in 12.30 PM (main bus station in Pula).

At the airport of Dubrovnik, many rent-a-car companies offer cars that can be driven to Montenegro.

Taxi service is also available from/to Dubrovnik Airport, and usual price is from 80 to 120 euros, depending from/to which Montenegrin town you want to go. (Example Kotor – 80 euros; Budva – 120 euros).