When I tried to start PC with new cmos battery the screen lit up for only a second and fans did not start at all.
Have you run the "paper clip" test yet with the power cables disconnected from the motherboard.
If not, we still don't know if the PSU is "working" or not. It could be on the point of catching fire.
If you read through this post on Tom's, you'll see the AGW650 is regarded as junk.
https://forums.tomshardware.com/threads/anyone-got-info-on-the-aresgame-650w-power-supply.3674079/
You're lucky to have got 5 years of use out of it. Replace it
now before it explodes. It may already have damaged your motherboard beyond repair.
After you replace cmos battery does the date and time have to be set manually or does it automatically detect them?
As I've already explained, you can set the time manually in the BIOS after replacing the battery (best option) or connect the computer to the internet and wait until Windows loads the correct date from a time server.
You could also update the time/date manually as soon as Windows starts.
Some programs throw up errors when the date reverts back to some distant time, several years ago in the past, when the CR2023 battery dies.
If you're lucky, when you buy a brand new (high quality) PSU, your computer will still work. Fingers crossed. If you're unlucky, you might have to replace the motherboard, if the Aresgame has destroyed it.
There are numerous "PSU Tier Lists" but try this one:
https://cultists.network/140/psu-tier-list/
If you choose something in Tier A (High-end), Tier B (Mid-range) or Tier C (Low-end) you stand a good chance of getting a decent PSU.
Please don't buy anything in Tier E (Avoid) or Tier F (Replace Immediately). It's just asking for more trouble.
How do I determine what size PS I need?
Provide us with a complete list of your PC components. Then we can advise on a new PSU.
At the very least we need to know the
exact model (part numbers) of the CPU, motherboard and GPU card (if fitted). The motherboard part number is usually printed somewhere near the PCIe slots (up in the corner).
For a low end PC, another (good quality) 650W PSU should be fine. For a more powerful PC, you might need 750W or higher, but if your existing system has been running on a flaky Aresgame 650W PSU, I doubt it pulls more than 250W.
Good luck.