Hi

Congratulations on the new client!
I don't work on retainer but as I understand it you bill monthly in advance for a specific number of hours. Most VAs who do work on retainer will have clients commit to contracts of an extended period of time (for instance 3, 6, 9, or 12 months).
Since you're starting in the middle of the month I think you could prorate it - that makes the most sense to me...
Hopefully a VA who DOES work on retainer can jump in here and fine tune my answer