Are you sure that you will get USB 3.0 speeds with such a card in anything less than a 16x PCIe slot? Chances are very high that your dual core processor is stuck on a time-period related motherboard. Which will have support for the PCIe 1.0 standard. That is a serious handicap for getting the maximum speed out of those USB 3.0 ports. And there is another thing to consider, even if your motherboard has enough PCIe "lanes" available, how sure are you if your CPU has support for all of those? And if the CPU has support, what other devices are connected to those lanes?
These are all specifications that the manufacturers of your motherboard and CPU provide on their web-pages.
And even if they can connect initially with USB 3.0 speeds, can the computer/CPU maintain those speeds? Dual core processors are not a good fit in this regard. Such processors are already busy enough with the workload of your processes and whatever the operating system needs to do in the background. If your computer had an quad core CPU in it, I would have much less reservations. You are likely able to connect the card into your computer, but it will behave barely better than USB 2.0. USB devices do negotiate their maximum transmission speeds, so the box of the expansion card may say USB 3.0, in reality you should be happy if your USB devices can manage USB.20 speeds (sustained).
Nothing mean is intended. A computer like yours is still running in my network, but only as a bare metal Linux server without any GUI. In such a setup, computers like yours are still pretty usable for light server duties that can take their time. I use it to create daily incremental and differential backups from 12 other servers and 17 workstations (using Bacula). And for that purpose a machine like yours is still of any use.
For 250 USD you should be able to get a much more modern motherboard (with 2 to 4 actually working USB 3.0 (or better) ports), support for the PCIe 3.0 standard or better, a 10th generation Intel CPU and 8 to 16GByte of DDR4 RAM. Windows 7, but also Windows 10 or Windows 11 will work perfectly fine on such a machine. You can still use your old case, monitor expansion cards, keyboard and mouse, if you so desire. But by changing the innards of your system, you will have actual hardware progress. A more worthy investment of money than an expansion card that may or may not work (well) in your current hardware. Your current computer may still be of use to you, but it is getting old. While the CPU in your machine will manage quite a few more years of service, I'm not so sure about the capacitors on your motherboard. Once those start to "pop", it will be over very quickly.
A mid-range (or better) motherboard pretty easily lasts 5 to 10 years without risks, if it was always connected to a high quality power supply, was never overclocked, always in a good power grid and periodically cleaned properly (dust, hair, lint, etc). Use it longer and you are essentially are starting to gamble with your data.
