I couldn't find the button that says 'try now'
If you've already clicked the "Try Now" button once, then it won't show up any more, but at that point you should have access to the CSS option. Look for it on the bottom of the menu on the right when you go to Appearance > Customize in your dashboard.
Does the purchase/upgrade make significant improvement?
The Custom Design upgrade let's you add custom CSS to modify the look of existing elements on a page. It doesn't let you change the HTML or make functional changes such as adding dynamic content in places the theme you're using doesn't already provide it.
And if I move to another wordpress.com theme do I pay again for another upgrade?
Nope. The Custom Design upgrade is applied per blog. You can change themes as many times as you want and still have access to the Custom Design upgrade as long as you're using the same blog.
You'll want to keep in mind that CSS is theme-specific though. What that means is that if you change themes, the old CSS you wrote will get saved in a revision because CSS written for one theme won't work the same way when applied to another. Each theme would need to have it's own custom CSS written separately.
I do think what you've asked is doable here at WordPress.com, but it will take some work and there are indeed some limitations compared to using a WordPress.org setup instead. The main limitation is that you cannot add adjust any theme's HTML or PHP code directly here at WordPress.com but you can get full control of all the code and manage and maintain that code on your own if you switch to a WordPress.org setup. Another key difference is support is done on for WordPress.org setups in volunteer based forums.
Try the example I posted again, but this time go straight to Appearance > Customize > CSS to apply it in a live preview. Hopefully it will give you an idea of what's possible, and if you still have questions after that just post them here and I'll do my best to help answer all of your questions!