The cheapest way is to take the bus from Dubrovnik to Herceg Novi (50km) at 9:30am or 3:30 pm and then continue from Herceg Novi to Podgorica.(you can catch the bus from there every half an hour).Distance from Herceg Novi to Podgorica is around 110 km.Whole trip should last around 4,5-5 hours.(depends on the key on border).Total amount is around 25 eur.
Also I can arrange you transfer for 140 eur/per car.
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