Either you rent a car or go by bus.
I'm sure Terminal tres Cruces sells tickets to Punta.
You can do both at the airport in Carrasco.
You don't say if you're arriving by ferry from Buenos aires or the airport from somewhere else, either way you can get a bus.
Bus from the airport, central terminal (Tres Cruces) or the port. Costs only about USD 7. Main operator is COT. You will get a ticket with a seat number on it, bags go in the hold. Takes about 90 mins.
Taxi is well over 100 USD, otherwise car hire
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