Is This Older PC Worth Upgrading, or Should I Build a New One?

If you are comfortable using alternatives to Microsoft Windows, installing a lightweight Linux distribution could give the existing computer a useful second life. It would receive current security updates and handle web browsing and office-style work reasonably well. Before doing that, though, I would test the old hardware carefully because shipping, failing drives, and aging power supplies can cause instability.

The storage drives are probably the only components worth considering, although their age means you should check their health and avoid relying on them as the only copy of important files. The case might also be reusable if it is in good condition, but the motherboard, processor, memory, and graphics card are on an obsolete platform with little practical upgrade potential.

I would not spend much money upgrading the old system. You could potentially reuse the case or drives, but buying replacement parts for that platform would put money into hardware that is already outdated. A new entry-level desktop or small-form-factor computer would be simpler, more reliable, and easily powerful enough for your stated use.

For browsing and writing, you do not need anything close to an $800 gaming PC. A modest modern system with around 16GB of memory and an inexpensive processor with integrated graphics would be more than enough. A budget build based on something like a Ryzen 5 5600G and a compatible motherboard would feel much smoother and give you a supported operating system, while still leaving plenty of room in the budget.