I have a shirt that's too long, but I have no time to sew it or bring it to a shop. I don't like tucking it in, because it doesn't look nice. How do I make it shorter without cutting anything, and make it seem like I cut it shorter?
Exchange it for a shorter one
Wash and tumble dry it on the hottest settings to shrink it; all my t-shirts shrink more vertically than they do horizontally
Use some hemming web, a heat based glue tape, to hem the t shirt shorter - check YouTube for tutorials on how to use it (faster than sewing)
Give it to a shop or relative who has the time to shorten it via the usual cut/hem method (sorry, some questions don't have a life hack, and there isn't a popular alternative to doing it properly because the proper way gives the best results for the least effort)
The answer to your question depends on if you're a man or a woman. If you're a man I don't think I can help you. If you're a woman, I have some ideas.
I don't know how many inches you have to work with, but you did say that it's a cotton shirt and that is great for these solutions. They shouldn't take long.
If you have at least three inches to cut, you could cut off the hem and cut fringe into the bottom of the shirt.
If you want you could also tie knots in the fringe, like this.
Another option would be to tie a knot in your shirt hem like these examples.