cold air in or hot air out?

We use a window fan, so in the part of the late afternoon when the sun is hitting our room, we use the fan on exhaust to pull hot air out of that room. Then once the sun moves and the evening air starts to cool down, we switch the fan to intake.

In a situation where you're moving air from one room to another, I'd probably try the same thing you suggested, to pull cooler air into that room. In a previous home, we used to have an air conditioner in one upstairs room but not the one directly across the hall due to the window set-ups in each room. And we'd leave the door to the room with the a/c open, and put a fan in the doorway of the cooler room to move cool air towards the open door of the warmer room. And it always seemed to help. I don't know if it would work in your situation, though.

I had the same setup in my California house.  Very high vaulted ceiling with zero insulation and massive windows causing the room to bake like an oven all summer.  When I needed the old duct work replaced due to asbestos, I had them put in a larger return into the master bedroom to get more air conditioned air in there.  It still didn’t work and I had to fall asleep downstairs on the family room couch during the summer, coming upstairs around 2 am.

Then I bought two heavy duty fans.  I put one in the window to pull the hot air out and put one in the doorway to push the cold air into the room.  It was better, but still not great.  I finally bought a portable room air conditioner the I had set up near the window with the exhaust hose blowing the hot air out while the chilled air wafted across my bed.  That ended up working the best for me.

In this new house, I cannot believe how even the temp stays all summer.  I have only had to run the air conditioner a handful of times when I was hauling boxes and climbing up on the stepladder to put things away.  If it was a light work day, I didn’t need to run it at all.  I think it helps that there is a crawl space below the house and full attic above the whole house that keeps the house quite livable.  And a previous owner had replaced all windows with double paned glass, which keeps the heat out even more.