There is plenty of public transport from Barnaul. You can take a direct bus to Gorno Altaisk (as we did 2012 with bicycles) or take the train to Bijsk and then the bus. From Gorno Altaisk there exist busses further on, but Kosh Agatsh has only few connections (this is as far as public transport gets you), it is advaisable to stop in between, as Altai is really beautiful.
From Kosh Agatsh you will have to find a taxi or hitchhike to Tashanta. In Tashanta in the morning you may well be able to get a lift on a truck (border usually opens at 9).
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