First few days on 20 mg of prozac (fluoxetine) cause drowsiness (i.e., feeling sleepy during the day despite 8-9 hours of sleep). could this subside eventually?

Yes: Often side effects go away after a few days or weeks. But - you might try adjusting when you take it. Early or late evening might work for you.

Side effects usually: Subside in 1-3 weeks. If you take the medication in the morning and feel sleepy during the day, try taking it at night and see if that reduces the daytime sleepiness. If it does not subside your prescriber may reduce the dose for a while and increase later when the side effect subsides. If it becomes intolerable and does not subside then an alternative medication may be necessary. Best regards.