If I arrive in the early morning at an airport for a late afternoon or evening American Airlines flight, and purchase a day pass for the Admirals Club, how many hours before the flight will they let me in?
There's no time limit - as long as you're traveling that day you will be allowed through security and allowed into the lounge. You don't even need to be flying AA - as long as you can physically get to the lounge and have paid your money then you will be allowed in.
The only real restriction is if you are planning to check luggage, in which case you will not be able to check your luggage until ~3-4 hours before the departure time of the flight.
I was turned away for showing up too early at the Admiral's Club in Buenos Aires (7 hrs prior to boarding of a 10 pm flight). Maybe the policy varies from club to club, but I was told 4 hours max, so come back later.
EDIT: I was totally wrong. I mistakenly went to the Tsar Alliance lounge when I meant to go to Admirals Club. The issue has been clarified (sitting in the AA club now). When I asked, AA clarified their policy. It's a true "day" pass so it should not matter how early you show up. I was also told it is good at the destination airport if you travel overnight.