worth to upgrade then

5600X3D is coming out soon no?

"What specs what you need to get 240 fps on average? (no recording)"

With no recording, anything like a Intel i5-12400F and a 6700XT, with 16 GB of 3200MHZ RAM should easily get you over 240 fps, around 320-340 on average.

"Would $1500 be a reasonable amount to spend to get 240fps + 240hz monitor? What about $2000? (Pre-Built)"

You would need around 1200$ to get that, even cheaper if you don't mind not having great looks. I have made a build for you to start off with, something I recommend, with upgrades being possible in the future
https://pcpartpicker.com/list/TLh6v3
PCPartPicker is a great website to find parts to build a PC doing it yourself, and the parts in the build above should be good as long as you already have a keyboard, mouse, etc.

"Is it a big enough difference going from 120fps - 240 to buy a new gaming PC?"
meh, worth if you expect to keep playing for a couple years

"Anything else I should know before buying a new PC?"
stick with a Intel CPU and AMD GPU for a build. They usually have the better price to performance compared to other companies. also for the people who say a RTX GPU is better, pc part market is shit, and Nvidia RTX cards have issues with high latency and input delay. also for optimizing pcs, check out this guide:
https://github.com/amitxv/PC-Tuning
also to build a pc all u need is a screwdriver (and common sense), happy building