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I visited Ushuaia last week.
There is no public transport between the town and the national park.
However, there are private shuttles that run between the central bus station and the national park, as well as a few other destinations.
Tickets are sold at a desk at the central bus station. They leave at three times in the morning, come back at three times in the afternoon. They can drop you off and pick you up from a few places in the park. The cost for a round trip is the equivalent of around 15 euros.