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You can find the requirement for Ubuntu 16.04.1 LTS from the above link. Your processor would be possible to run it, but not that feasible. Upgrading your hardware might be much better than waste time in vain. Old versions of Ubuntu might be designed for your systems, but phased-out long ago.
As to single core Athlon 64 processor, the most potential version of Windows is Windows 8 (not 8.1) x64. But you need at least 1GB system memory and Direct 9c level GPU. Ubuntu is different from Windows, you would never expect its performance and usage better than Windows.
I wouldn't recommend older versions of Ubuntu (ones that are still available) as being more suitable as the CPU & RAM requirements have overall changed very little compared to the current versions. There are flavours of Ubuntu (e.g. Xubuntu and Lubuntu) that cater for lower-spec hardware, and as these are all current builds they don't compromise on bug fixes, compatibility or support. These may provide a better desktop experience than contemporary Windows on the same hardware, but overall the OP's needs are as you stated probably better served by more recent hardware, whatever the OS.
I have been running various versions of Ubuntu for many years on AMD processors from the original Athlon to the FX8320e. I got to Ubuntu 14.04 LTS (32-bit) on an Athlon XP 2200+ before ditching that CPU as it could no longer give adequate performance (even with 1.5G RAM), and was limited to 32-bit operation.
A single core Athlon 64 is a little better so can run Ubuntu 16.04 if you have at least 2G RAM and a GPU with supported drivers, but the motherboard for that age of CPU is unlikely to support SATA II or better, so an SSD upgrade won't be of much help. You'll find that such a system will function OK for basic desktop use, but will often chug away on disk I/O especially when starting things up. The default desktop environment in Ubuntu (Unity) is quite demanding even with the recent low graphics mode in 16.04, so you can try the Lubuntu or Xubuntu flavours for less demand on the CPU and RAM.
Overall though, if you don't have any specific legacy support requirements (e.g. specialist ISA/PCI cards or peripherals that need old ports) you're better off with upgrading to a newer platform - second hand parts are cheap and plentiful.