The Black Rock CRASHED onto the Island at nighttime. Old sailing ships like the Black Rock weren't all that fast, so it would've taken a while for the Black Rock to actually arrive at the Island.
Also, we know that time behaves differently on and around the Island. The difference in time seems to depend on where/when the Island is currently located. Therefore, it may also be that it was night in the real world, and day on the Island.
Remember, it is already daytime when Richard wakes up. It is possible that that wasn't because of time passing, and was instead caused by time distortion.