This might work for some people but for me, if by 1AM, I'm not sleepy and I get out of bed to do something else, there's probably no chance I'll sleep at any reasonable time and then be able to wake up at a time like 7AM. I would in fact probably sleep in that day and ruin my schedule, which could start a negative trend. From my experience, I find that even just lying in bed and resting my eyes even if I can't fully sleep is the best way to go. A few hours of meaningful rest in bed alone with my thoughts is time well spent. 1AM is too late to start "occupying myself until my body is ready to sleep". If I needed to do something like this, I would start a few hours before the time I'd like to sleep. First tiring myself out with a quick workout or something and then taking a cold shower which will make me want to bundle up inside a blanket and get super comfortable.
Yes, working out and showers are definitely good for your sleep.
While it may be hard to wake up at 7, if you could do it, it would generate what we call 'sleep pressure' for the next night. Basically you are tiring your self into sleeping more when you are in bed vs being awake and tossing and turning.
Also when you hop out of bed, do something relaxing like listening to music or reading a book. If you started working it could stimulate your mind and keep you up even longer.
