If you're injecting an anesthetic into the sub-arachnoid space, you're injecting it into the CSF since that's where the CSF is. I think it should get distributed throughout the CSF.
The blood brain barrier shouldn't come into play here since we're not injecting the anesthetic into the blood. The barrier of concern here would be the blood CSF barrier. I'm not sure if anesthetics can cross this barrier and what the consequences of that would be.