If you are going to use a cube for No Chilling, I just don't see any reason not to follow the vetted No Chill doctrine of the Aussies, which says, very clearly "Do not be tempted to cool the cube prematurely". Hundreds of Aussie brewers pioneered this technique, and its "common knowledge" in that group that you can mess up and catch an infection in the cube if you mess with the hot tank before its near room temp. Personally, I No Chilled something like 40+ batches and my first failure came from dropping my wort down to 180F BEFORE I drained it into the cube. I'm not stating definitively that that was the cause, but it echos a lot of other No Chillers experiences.
Technically, pasteurization of the flat surfaces inside the cube happens within 15minute or so, but we aren't just pasteurizing the liquid itself or the surface of the cube; you are also waiting for the inner rings of the screw-on cap to heat up as well, which doesn't happen instantly as plastic isn't super conductive.
Summary: I'd give the tank at least an hour from the time you fill it before you start to chill it if you are in an absolute rush, but it would be preferable to wait overnight, per the "standard" No Chill practice. And make sure you turn the tank so that it's on its side so that the cap is submerged.
If you want to forgo the No Chill practice, then you can do the following to chill your brew without an imersion chiller:
1) Buy a large plastic tub from Lowes (should run less than $10) and maybe 30lbs of ice (look for those automated dispensers, they run like $2 per 20lbs now I think). Try to find a tub with a spigot at the bottom if you can.
2) Put in a few gallons of water, and dedicated piece of fabric, like an old towel or a patch of denim at the bottom of the tub
3) Right after the boil, lower your kettle into the tub onto the fabric so that the bottom of the kettle doesn't scorch the plastic. Don't worry too much about this though. Without additional heat, the kettle will very quickly cool to the temp of the wort (212F) which isn't hot enough by itself to melt the plastic.
4) Swirl the water in the tub around the base of the kettle clockwise, while you use a brewing spoon to stir the wort counter-clockwise. Spinning the wort clockwise and the water counter-clockwise will give your beer the flavor of Satan's buttcrack (ok I made that last part up, just make sure you are stirring them in opposite directions)
5) The water will very quickly heat up. Drain off that water and save it for cleaning your kettle later. Add some new water, stir it for a few min, then add in the ice to the water, but not to your wort. Don't try to put all the ice in at once, just get the level of the ice up near the rim of the kettle, but not over it. Resume the stirring.
6) If you constantly stir the ice-slush and wort in opposite directions, your wort will drop to pitching temps in 20min maybe. Once you get around 100F, I'd suggest transfering it into your carboy, then putting that back into the ice-slush. You can leave it there for quite a while by itself, or keep stirring to get it down ASAP. When the ice nearly all melts, just drain a bunch of water out and add more ice.
Other notes: its better to have more ice than you need and just cost a few extra bucks. Also, you can plan to boil down less than 5gal, and have some jugs of pre-boiled water sitting in your fridge ready to be added as top-off water at 32F. A gallon of 32F water can drop 4gal of wort at 75F to 5gal at 65F, in my experience.