1 Hour is more than enough to seriously threaten the entire basis of the Japanese attack. The Fleet will have water tight doors closed, most of its anti aircraft weapons manned, damage control teams assembled, and some of the crews who are ashore (especially a lot of the officers at home with their families) have time to get to their ships.
note this map
http://history-world.org/pearlmap2.jpg
the most serious damage to the fleet was from the torpedo bombers, which sank the California, West Virginia and Oklahoma, and put torpedoes into the Arizona. This damage was done in the initial few minutes as the torpedo bombers were unopposed. With an hours warning which assumes an alert triggered by the Aaron Wards engagement with a submarine and the radar warning taken seriously, which assumes a few minutes delay of the air defense staff going to breakfast... both of which could have been set in motion by Marshall's staff sending a radio message or or more urgently marked message (it was not marked high urgency).
So note the flight paths of the torpedo planes, and note they fly right over the cruisers which would have a really good chance of shooting the hell out of them. In OTL the Conyingham managed to shoot 3 torpedo planes, in spite of surprise, by 8:08 AM (first 13 minutes of the attack). Consider how vulnerable the rest of them would have been if surprise was lost.
A few fighters might be enough to disrupt the level bombing attack, which could possibly save the Arizona..
http://www.amazon.com/Attack-Pearl-Harbor-Strategy-Deceptions/dp/1612001971
this book analyzed thoroughly the attack, and estimates warning doubles or more the Japanese losses, and results in far less serious damage... half or fewer torpedo hits and bomb hits. Fewer hits and damage control means almost certainly the Oklahoma does not roll over, and most likely the West Virginia is far less damaged. The California is still screwed though, it needs more time to secure hatches that were removed for the inspection planned.
The Japanese did not at any point prior to the attack seriously consider attacking the oil and base facilities. An attack of opportunity against the facilities was only considered after success in the first two strikes.
If the fleet has steamed out of the harbor, the Japanese would go looking for it instead of hitting the fleet base. It is why they sent float planes out first that morning, checking not only Pearl but other likely anchorages as well.
Zimm also points out, using data from Midway and the Solomons campaign, that the fleet at sea is not as vulnerable as has been suggested, pointing out that even 1941 US Navy air defense is pretty powerful for the time period, the level bombers would be unlikely to cause serious damage to moving targets, and the dive bombers lacked bombs powerful enough to penetrate battleship deck armor (the Arizona was hit by a level bomber with a converted artillery shell, the Nevada was beached because of its torpedo damage, not because of deck damage caused by the dive bombers). That fleet operating in Hawaiian waters would also have fighter cover, and while American pilots were not prepared for the Zero, there are a lot of them, and breaking up an attack and reducing its effectiveness is more important than gaining kills and worth severe losses.
So it seems likely based on this that an hour is really decisive, and a specific warning to prepare not only for internal sabotage but possible external attack (which TWO US Navy fleet problems in the 1930s showed was possible) would potentially get the Army air defense batteries deployed to their air defense positions in time to make a huge difference as well.
The USN had excellent AAA for the period - Bofors, plus 5" DP, plus radars and analog computers is lethal. But the Standards aren't up to that level until after they are rebuilt. On 12/7/1941, the most valuable part of them are their crews. Now, the can probably beat the level bombers in open waters - but can they beat the torpedo bombers, if those bombers have an escort? If they can't - and it's likely that several of them don't - that's several thousand men right there. A potential surface fight at that stage of the war doesn't bear thinking about. Think Savo Island - with even less experience with the radar if contact occurs at night.