If you take highway 1 the whole way it would take 10 to 12 hours.
You could take 101 to Paso Robles, that should take about 4, 4 1/2 hours from DLR.
At Paso Robles you can take 46 to the coast and get on highway 1 north. Paso Robles to the Monterey Bay Aquarium should take 3 or 4 hours without any stops but you'll want to stop and admire the views.
At the 46 highway 1 interchange you'll pass through the cute town of Cambria and then San Simeon where Hearst Castle is. Just a bit north of San Simeon is Pt. Piedras Blancas, it's an elephant seal rookery and is a must see on the coast. After that you'll wind your way up the through the Big Sur Coast.
You can also take 101 all the way to 68 which will take you to Monterey. This route should only take 6 1/2 hours with no traffic. You'll still get some coastal views too, but nothing like Big Sur.