Mestia is not a large town and can be walked out of easily. It has the major Svaneti road running through it. It also has a road that splits off to the north toward the border with Russia and a trekking trail to the Lekhziri glacier.
Mestia can easily be walked out of to get onto the Svaneti road. If you watch the road markers you can make sure you are heading the right direction. The markers are skinny white poles with two square blue plaques declaring the kilometer distances to the ends of the road. If the number facing you is around 140 then you are heading to Ushguli and Lentekhi. If it is around 65 then you are heading to Zugdidi.
No information for hitching to Ushguli is available at the time of writing.