Is there a way to get from Podgorica airport to Budva by public transportation?
You can take the train to Bar from the "Aerodrom" train station, which is about one kilometer from Podgorica Airport (as you reach an elevated bridge, don't cross it, but keep walking to the left, and you'll reach the platform). There is approximately one train per hour. Then from Bar you'll have buses going to Budva.
Otherwise, if you can get to central Podgorica, there are direct buses to Budva from the bus station.