The most sensible way to get from Buenos Aires to La Paloma (Uruguay) is via ferry or bus though it won't bne direct. You could catch a ferry from Buenos Aires to Montevideo and then catching a bus up to La Paloma, or you could take Buquebus from Buenos Aires directly to Montevideo. Once there, take a cab to Tres Cruces bus terminal and from there a bus to La Paloma.
Get the ferry to Montevideo or Colonia, hire a car & drive to La Paloma as roads are very good. From Colonia to La Paloma is about 4 hours drive. From Montevideo is only 2 hours. You could also fly to Punta del Este, and rent a car to get to La Paloma.